THE  THEORY  OF  MACHINES 


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THE 
THEORY  OF  MACHINES 

f 
PART  I 

THE  PRINCIPLES  OF  MECHANISM 

PART  II 
ELEMENTARY  MECHANICS  OF  MACHINES 


BY 
ROBERT  W.  ANGUS,  B.A.Sc., 

MEMBER    OF   THE    AMERICAN   SOCIETY   OF   MECHANICAL   ENGINEERS, 

PROFESSOR    OF    MECHANICAL   ENGINEERING,   UNIVERSITY    OF 

TORONTO,  TORONTO,  CANADA 


SECOND  EDITION 
SECOND  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC, 

239  WEST  39TH  STREET.     NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 
6  &  8  BOUVERIE  ST.,  E.  C. 

1917 


COPYRIGHT,  1917,  BY  THE 
McGRAw-HiLL  BOOK  COMPANY,  INC. 


THE     MAl'l.K     I'KKSS     YORK     PA 


PREFACE 

The  present  treatise  dealing  with  the  Principles  of  Mechanism 
and  Mechanics  of  Machinery  is  the  result  of  a  number  of  years' 
experience  in  teaching  the  subjects  and  in  practising  engineering, 
and  endeavors  to  deal  with  problems  of  fairly  common  occur- 
rence. It  is  intended  to  cover  the  needs  of  the  beginner  in  the 
study  of  the  science  of  machinery,  and  also  to  take  up  a  number 
of  the  advanced  problems  in  mechanics. 

As  the  engineer  uses  the  drafting  board  very  freely  in  the 
solution  of  his  problems,  the  author  has  devised  graphical  solu- 
tions throughout,  and  only  in  a  very  few  instances  has  he  used 
formulae  involving  anything  more  than  elementary  trigonometry 
and  algebra.  The  two  or  three  cases  involving  the  calculus  may 
be  omitted  without  detracting  much  from  the  usefulness  of  the 
book. 

The  reader  must  remember  that  the  book  does  not  deal  with 
machine  design,  -and  as  the  drawings  have  been  made  for  the 
special  purpose  of  illustrating  the  principles  under  discussion, 
the  mechanical  details  have  frequently  been  omitted,  and  in  cer- 
tain cases  the  proportions  somewhat  modified  so  as  to  make  the 
constructions  employed  clearer. 

The  phorograph  of  Professor  Rosebrugh  has  been  introduced 
in  Chapter  IV,  and  appeared  in  the  first  edition  for  the  first  time 
in  print.  It  has  been  very  freely  used  throughout,  so  that  most 
of  the  solutions  are  new,  and  experience  has  shown  that  results 
are  more  easily  obtained  in  this  way  than  by  the  usual  methods. 

As  the  second  part  of  the  book  is  much  more  difficult  than  the 
first,  it  is  recommended  that  in  teaching  the  subject  most  of  the 
first  part  be  given  to  students  in  the  sophomore  year,  all  of  the 
second  part  and  possibly  some  of  the  first  part  being  assigned  in 
the  junior  year. 

The  thanks  of  the  author  are  due  to  Mr.  J.  H.  Parkin  for  his 
careful  work  on  governor  problems,  some  of  which  are  incorpor- 
ated, and  for  assistance  in  proofreading;  also  to  the  various  firms 
and  others  who  furnished  cuts  and  information,  most  of  which 
is  acknowledged  in  the  body  of  the  book. 

V 


404993 


vi  PREFACE 

The  present  edition  has  been  entirely  rewritten  and  enlarged 
and  all  of  the  previous  examples  carefully  checked  and  corrected 
where  necessary.  The  cuts  have  been  re-drawn  and  many  new 
ones  added;  further,  the  Chapter  on  Balancing  is  new.  Ques- 
tions at  the  end  of  each  chapter  have  been  added. 

R.  W.  A. 

UNIVERSITY  OP  TORONTO, 
February,  1917. 


CONTENTS 

PAGE 

PREFACE v 

SYMBOLS  USED xi 

PART  I 

THE  PRINCIPLES  OF  MECHANISM 
CHAPTER  I 

THE  NATURE  OF  THE  MACHINE 3 

General  discussion — Parts  and  purpose  of  the  machine — Definitions 
— Divisions  of  the  subject — Constrained  motion — Turning  and 
sliding  motion — Mechanisms — Inversion  of  the  chain — Examples 
— Sections  1  to  26. 

CHAPTER  II 

MOTION  IN  MACHINES 24 

Plane  motion — Data  necessary  to  locate  a  body  and  describe  its 
motion — Absolute  and  relative  motion — The  virtual  center — Fixed 
and  permanent  centers — Location  of  the  virtual  center — Sections 
27  to  41. 

CHAPTER  III 

VELOCITY  DIAGRAMS 35 

Application  of  virtual  center — Linear  and  angular  velocities — Appli- 
cation to  various  links  and  machines — Graphical  representation — 
Piston  velocity  diagrams — Pump  discharge— Sections  42  to  56. 

CHAPTER  IV 

THE  MOTION  DIAGRAM 49 

Phorograph  method  of  determining  velocities — Fundamental  prin- 
ciples— Images  of  points — Application — Image  of  link  gives  angular 
velocity — Sense  of  rotation — Phorograph  a  vector  diagram — Steam 
engine — Whitworth  quick-return  motion — Valve  gears,  etc. — Sec- 
tions 57  to  80. 

CHAPTER  V 

TOOTHED  GEARING 68 

Forms  of  drives  used — Spur  gearing — Proper  outlines  and  condi- 
tions to  be  fulfilled — Cycloidal  teeth — Involute  teeth — Parts  and 

vii 


viii  CONTENTS 

PAGE 

proportions  of  teeth — Definitions — Racks — Internal  gears — Inter- 
ference— Stub  teeth — Module — Helical  teeth — Sections  81  to  103. 

CHAPTER  VI 

BEVEL  AND  SPIRAL  GEARING 90 

Various  types  of  such  gearing — Bevel  gearing— Teeth  of  bevel 
gears — Spiral  tooth  bevel  gears — Skew  bevel  gearing — Pitch  sur- 
faces— General  solution  of  the  problem — Applications — Screw 
gearing — Worm  and  worm-wheel  teeth — General  remarks — Sec- 
tions 104  to  125. 

CHAPTER  VII 

TRAINS  OF  GEARING. 110 

Kinds  of  gearing  trains — Ordinary  trains — Ratio — Idlers — Ex- 
amples— Automobile  gear  box — Screw-cutting  lathe — Special 
threads — Epicyclic  or  planetary  gearing — Ratio — Weston  triplex 
block — Drill,  etc. — Ford  transmission — Sections  126  to  139. 

CHAPTER  VIII 

CAMS 136 

Purpose  of  cams — Stamp  mill  cam — Uniform  velocity  cam — Design 
of  cam  for  shear — General  problem  of  design — Application  to  gas 
engine — Sections  140  to  144. 

CHAPTER  IX 

FORCES  ACTING  IN  MACHINES 149 

Classification  of  forces  acting  in  machines — Static  equilibrium — 
Solution  by  virtual  centers — Examples — Solution  by  phorograph — 
Shear — Rock  crusher — Riveters,  etc. — Sections  145  to  152. 

CHAPTER  X 

CRANK  EFFORT  AND  TURNING  MOMENT  DIAGRAMS 164 

Variations  in  available  energy — Crank  effort — Torque — Steam  en- 
gine— Crank  effort  from  indicator  diagrams — Types  of  engines — 
Internal  combustion  engines — Effect  of  various  arrangements — 
Sections  153  to  161. 

CHAPTER  XI 

THE  EFFICIENCY  OF  MACHINES 176 

Input  and  output — Meaning  of  efficiency — Methods  of  expressing 
efficiency — Friction — Friction  factor — Sliding  pairs — Turning 
Pairs — Complete  machines — Sections  162  to  174. 


CONTENTS  ix 

PART  II 

MECHANICS  OF  MACHINERY 
CHAPTER  XII 

PAGE 
GOVERNORS 201 

Methods  of  governing — Purpose  of  the  governor — Fly-ball 
governors  —  Powerfulness — Sensitiveness — Isochronism — A  c  t  u  a  1 
design — Characteristic  curves — Spring  governor — Inertia  gover- 
nor— Distribution  of  weight — Sections  175  to  198. 

CHAPTER  XIII 

SPEED  FLUCTUATIONS  IN   MACHINERY 240 

Cause  of  speed  fluctuations — Illustration  in  case  of  steam  engine 
— Kinetic  energy  of  machines  and  bodies — Reduced  inertia  of  bodies 
and  machines — Speed  fluctuations — Graphical  determination  for 
given  machine — Practical  application  in  a  given  case — Sections 
199  to  213. 

CHAPTER  XIV 

THE  PROPER  WEIGHT  OF   FLYWHEELS 261 

Purpose  of  flywheels — Discussion  of  Methods — Dimensions  of 
wheels — Coefficient  of  speed  fluctuation — The  E-J  diagram — 
Numerical  examples  on  several  machines — Sections  214  to  223. 

CHAPTER  XV 

ACCELERATIONS  IN  MACHINERY  AND  THEIR  EFFECTS 277 

General  effects  of  acceleration — Normal  and  tangential  accelera- 
tion— Graphical  construction — Machines — Disturbing  forces — 
Stresses  due  to  inertia — Examples — Numerical  problem  on  an  en- 
gine— Sections  224  to  243. 

CHAPTER  XVI 

BALANCING  OF  MACHINERY 307 

Discussion  on  balancing — Balancing  of  rotating  masses — Single 
mass — Several  masses — Reciprocating  and  swinging  masses — 
Primary  balancing — Secondary  balancing — Short  connecting  rod 
— Four  crank  engine — Locomotive — Motor  cycle  engine — Sections 
244  to  254. 

APPENDIX  A:  FORMULA  FOR  PISTON  ACCELERATION 329 

APPENDIX  B:  THE  MOMENT  OF   INERTIA  OF  A  BODY 331 

INDEX . .  .  333 


SYMBOLS  USED 

The  following  are  some  of  the  symbols  used  in  this  book,  with 
the  meanings  usually  attached  to  them. 
w  =  weight  in  pounds. 

g  =  acceleration  of  gravity  =  32.2  ft.    per  second  per 
second. 

w 
m  =  mass  =  - 


v  =  velocity  in  feet  per  second. 
n  =  revolutions  per  minute. 

co  =  radians  per  second  =  ~~^~ 


TT  =  3.1416. 

a  =  angular  acceleration  in  radians  per  second  per  second. 
0  =  crank  angle  from  inner  dead  center. 
/  =  moment  of  inertia  about  the  center  of  gravity. 
k  =  radius  of  gyration  in  feet  =  \/I/m 
J  =  reduced  inertia  referred  to  primary  link. 
T  =  torque  in  foot-pounds. 

P,  Pf,  P"  represent  the  point  P  and  its  images  on  the  velocity 
and  acceleration  diagrams  respectively. 


PART  I 
THE  PRINCIPLES  OF  MECHANISM 


THE  THEORY  OF  MACHINES 

CHAPTER  I 
THE  NATURE  OF  THE  MACHINE 

1.  General. — In  discussing  a  subject  it  is  important  to  know 
its  distinguishing  characteristics,  and  the  features  which  it  has 
in  common  with  other,  and  in  many  cases,  more  fundamental 
matters.     This  is  particularly  necessary  in  the  case  of  the  machine, 
for  the  problems   connected  with  the  mechanics  of  machinery 
do  not  differ  in  many  ways  from  similar  problems  in  the  mechan- 
ics of  free  bodies,  both  being  governed  by  the  same  general  laws, 
and  yet  there  are  certain  special  conditions  existing  in  machinery 
which  modify  to  some  extent  the  forces  acting,  and  these  condi- 
tions must  be  studied  and  classified  so  that  their  effect  may  be 
understood. 

Again,  machinery  has  recently  come  into  very  frequent  use, 
and  is  of  such  a  great  variety  and  number  of  forms,  that  it  de- 
serves special  study  and  consideration,  and  with  this  in  mind  it 
will  be  well  to  deal  with  the  subject  specifically,  applying  the 
known  laws  to  the  solution  of  such  problems  as  may  arise. 

2.  Nature  of  the  Machine. — In  order  that  the  special  nature 
of  the  machine  may  be  best  understood,  it  will  be  most  con- 
venient to  examine  in  detail  one  or  two  well-known  machines 
and  in  this  way  to  see  what  particular  properties  they  possess. 
One  of  the  most  common  and  best  known  machines  is  the  recip- 
rocating engine,  (whether  driven  by  steam  or  gas  is  unimportant) 
which   consists   of  the   following   essential,   independent   parts: 
(a)  The  part  which  is  rigidly  fixed  to  a  foundation  or  the  frame- 
work of  a  ship,  and  which  carries  the  cylinder,  the  crosshead 
guides,  if  these  are  used,  and  at  least  one  bearing  for  the  crank- 
shaft, these  all  forming  parts  of  the  one  rigid  piece,  which  is  for 
brevity  called  the  frame,  and  which  is  always  fixed  in  position. 
(6)  The  piston,  piston  rod  and  crosshead,  which  are  also  parts 
of  one  rigid  piece,  either  made  up  of  several  parts  screwed  to- 
gether as  in  large  steam  and  gas  engines,  or  of  a  single  casting 

3 


4'  THE  THEORY  OF  MACHINES 

as  in  automobile  engines,  where  the  piston  rod  is  entirely  omitted 
and  the  crosshead  is  combined  with  the  piston.  It  will  be  con- 
venient to  refer  to  this  part  as  the  piston,  and  it  is  to  be  noticed 
that  the  piston  always  moves  relatively  to  the  frame  with  a 
motion  of  translation,1  and  further  always  contains  the  wristpin, 
a  round  pin  to  facilitate  connection  with  other  parts.  The  pis- 
ton then  moves  relatively  to  the  frame  and  is  so  constructed  as 
to  pair  with  other  parts  of  the  machine  such  as  the  frame  and 
connecting  rod  now  to  be  described,  (c)  The  connecting  rod 
is  the  third  part,  and  its  motion  is  peculiar  in  that  one  end  of  it 
describes  a  circle  while  the  other  end,  which  is  paired  with  the 
wristpin,  moves  in  a  straight  line,  which  latter  motion  is 
governed  by  the  piston.  All  points  on  the  rod  move  in  parallel 
planes,  however,  and  it  is  said  to  have  plane  motion,  as  has  also 
the  piston.  The  purpose  of  the  rod  is  to  transmit  the  motion  of 
the  piston,  in  a  modified  form,  to  the  remaining  part  of  the 
machine,  and  for  this  purpose  one  end  of  it  is  bored  out  to  fit 
the  wristpin  while  the  other  end  is  bored  out  to  fit  a  pin  on  the 
crank,  which  two  pins  are  thus  kept  a  fixed  distance  apart  and 
their  axes  are  always  kept  parallel  to  one  another,  (d)  The 
fourth  and  last  essential  part  is  the  crank  and  crankshaft,  or, 
as  it  may  be  briefly  called,  the  crank.  This  part  also  pairs  with 
two  of  the  other  parts  already  named,  the  frame  and  the  connect- 
ing rod,  the  crankshaft  fitting  into  the  bearing  arranged  for  it 
on  the  frame  and  the  crankpin,  which  travels  in  a  circle  about  the 
crankshaft,  fitting  into  the  bored  hole  in  the  connecting  rod 
available  for  it.  The  stroke  of  the  piston  depends  upon  the 
radius  of  the  crank  or  the  diameter  of  the  crankpin  circle,  and  is 
equal  to  the  latter  diameter  in  all  cases  where  the  direction  of 
motion  of  the  piston  passes  through  the  center  of  the  crankshaft. 
The  flywheel  forms  part  of  the  crank  and  crankshaft. 

In  many  engines  there  are  additional  parts  to  those  mentioned, 
steam  engines  having  a  valve  and  valve  gear,  as  also  do  many 
internal-combustion  engines,  and  yet  a  number  of  engines  have 
no  more  than  the  four  parts  mentioned,  so  that  these  appear  to 
be  the  only  essential  ones. 

3.  Lathe. — Another  well-known  machine  may  be  mentioned, 
namely,  the  lathe.  All  lathes  contain  a  fixed  part  or  frame  or 

1  By  a  motion  of  translation  is  meant  that  all  points  on  the  part  considered 
move  in  parallel  straight  lines  in  the  same  direction  and  sense  and  through 
the  same  distance. 


THE  NATURE  OF  THE  MACHINE        5 

bed  which  holds  the  fixed  or  tail  center,  and  which  also  contains 
bored  bearings  for  the  live  center  and  gearshafts.  Then  there  is 
the  live  center  which  rotates  in  the  bearings  in  the  frame  and  which 
drives  the  work,  being  itself  generally  operated  by  means  of  a 
belt  from  a  countershaft.  In  addition  to  these  parts  there  is  the 
carriage  which  holds  the  tool  post  and  has  a  sliding  motion  along 
the  frame,  the  gears,  the  lead  screw,  belts  and  other  parts,  all 
of  which  have  their  known  functions  to  perform,  the  details  of 
which  need  not  be  dwelt  upon. 

4.  Parts  of  the  Machine. — These  two  machines  are  typical  of 
a  very  large  number  and  from  them  the  definition  of  the  machine 
may  be  developed.     Each  of  these  machines  contains  more  than 
one  part,  and  in  thinking  of  any  other  machine  it  will  be  seen 
that  it  contains  at  least  two  parts :  thus  a  crowbar  is  not  a  machine, 
neither  is  a  shaft  nor  a  pulley;  if  they  were,  it  would  be  difficult 
to  conceive  of  anything  which  was  not  a  machine.     The  so-called 
" simple  machines,"  the  lever,  the  wheel  and  axle,  and  the  wedge 
cause  confusion  along  this  line  because  the  complete  machine 
is  not  inferred  from  the  name:  thus  the  bar  of  iron  cannot  be 
called  a  lever,  it  serves  such  a  purpose  only  when  along  with  it 
is  a  fulcrum;  the  wheel  and  axle  acts  as  a  machine  only  when  it 
is  mounted  in  a  frame  with  proper  bearings;  and  so  with  the 
wedge.     Thus  a  machine  consists  of  a  combination  of  parts. 

5.  Again,  these  parts  must  offer  some  resistance  to  change  of 
shape  to  be  of  any  value  in  this  connection.     Usually  the  parts 
of  a  machine  are  rigid,  but  very  frequently  belts  and  ropes  are 
used,  and  it  is  well  known  that  these  serve  their  proper  purpose 
only  when  they  are  in  tension,  because  only  when  they  are  used 
in  this  way  do  they  produce  motion  since  they  offer  resistance 
to  change  of  shape.     No  one  ever  puts  a  belt  in  a  machine  in  a 
place  where  it  is  in  compression.     Springs  are  often  used  as  in 
valve  gears  and  governors,  but  they  offer  resistance  wherever 
used.     Thus  the  parts  of  a  machine  must  be  resistant. 

6.  Relative  Motion. — Now  under  the  preceding  limitations  a 
ship  or  building  or  any  other  structure  could  readily  be  included, 
and  yet  they  are  not  called  machines,  in  fact  nothing  is  a  machine 
in  which  the  parts  are  incapable  of  motion  with  regard  to  one 
another.     In  the  engine,  if  the  frame  is  stationary,  all  the  other 
parts  are  capable  of  moving,  and  when  the  machine  is  serving 
its  true  purpose  they  do  move; 'in  a  bicycle,  the  wheels,  chain, 
pedals,  etc.,  all  move  relatively  to  one  another,  and  in  all  machines 


6  THE  THEORY  OF  MACHINES 

the  parts  must  have  relative  motion.  It  is  to  be  borne  in  mind 
that  all  the  parts  do  not  necessarily  move,  and  as  a  matter  of 
fact  there  are  very  few  machines  in  which  one  part,  which  is 
referred  to  briefly  as  the  frame,  is  not  stationary,  but  all  parts 
must  move  relatively  to  one  another.  If  one  stood  on  the  frame 
of  an  engine  the  motion  of  the  connecting  rod  would  be  quite 
evident  if  slow  enough;  and  if,  on  the  other  hand,  one  clung  to 
the  connecting  rod  of  a  very  slow-moving  engine  the  frame  would 
appear  to  move,  that  is,  the  frame  has  a  motion  relative  to  the 
connecting  rod,  and  vice  versa. 

7.  In  a  bicycle  all  parts  move  when  it  is  going  along  a  road, 
but  still  the  different  parts  have  relative  motion,   some  parts 
moving  faster  than  others,  and  in  this  and  in  many  other  similar 
cases,  the  frame  is  the  part  on  which  the  rider  is  and  which  has 
no  motion  relative  to  him.     In  case  of  a  car  skidding  down  a 
hill,  all  parts  have  exactly  the  same  motion,  none  of  the  parts 
having  relative  motion,  the  whole  acting  as  a  solid  body. 

8.  Constrained  Motion. — Now  considering  the  nature  of  the 
motion,  this  also  distinguishes  the  machine.     When  a  body  moves 
in  space  its  direction,  sense  and  velocity  depend  entirely  upon  the 
forces  acting  on  it  for  the  time  being,  the  path  of  a  rifle  ball 
depends  upon  the  force  of  the  wind,  the  attraction  of  gravity, 
etc.,  and  it  is  impossible  to  make  two  of  them  travel  over  exactly 
the  same   path,   because  the  forces  acting  continually  vary;  a 
thrown  ball  may  go  in  an  approximately  straight  line  until  struck 
by  the  batter  when  its  course  suddenly  changes,  so  also  with  a  ship, 
that  is,  in  general,  the  path  of  a  free   body  varies  with  the 
external  forces  acting  upon  it.     In  the  case  of  the  machine,  how- 
ever, the  matter  is  entirely  different,  for  the  path  of  each  part  is 
predetermined  by  the  designer,  and  he  arranges  the  whole  machine 
so  that  each  part  shall  act  in  conjunction  with  the  others  to 
produce  in  each  a  perfectly  defined  path. 

Thus,  in  a  steam  engine  the  piston  moves  in  a  straight  line 
back  and  forth  without  turning  at  all,  the  crankpin  describes  a 
true  circle,  each  point  on  it  remaining  in  a  fixed  plane,  normal 
to  the  axis  of  the  crankshaft  during  the  rotation,  while  also  the 
motion  of  the  connecting  rod,  although  not  so  simple  is  perfectly 
definite.  In  judging  the  quality  of  the  workmanship  in  an 
engine  one  watches  to  see  how  exact  each  of  these  motions  is 
and  how  nearly  it  approaches  to  what  was  intended ;  for  example, 
if  a  point  on  the  crank  does  not  describe  a  true  circle  in  a  fixed 


THE  NATURE  OF  THE  MACHINE  1 

plane,  or  the  crosshead  does  not  move  in  a  perfectly  straight  line 
the  engine  is  not  regarded  as  a  good  one. 

The  same  general  principle  applies  to  a  lathe;  the  carriage 
must  slide  along  the  frame  in  an  exact  straight  line  and  the  spindle 
must  have  a  true  rotary  motion,  etc.,  and  the  lathe  in  which  these 
conditions  are  most  exactly  fulfilled  brings  the  highest  price. 

These  motions  are  fixed  by  the  designer  and  the  parts  are 
arranged  so  as  to  constrain  them  absolutely,  irrespective  of  the 
external  forces  acting;  if  one  presses  on  the  side  of  the  crosshead 
its  motion  is  unchanged,  and  if  sufficient  pressure  is  produced 
to  change  the  motion  the  machine  breaks  and  is  useless.  The 
carriage  of  the  lathe  can  move  only  along  the  frame  whether 
the  tool  which  it  carries  is  idle  or  subjected  to  considerable 
force  due  to  the  cutting  of  metal;  should  the  carriage  be  pushed 
aside  so  that  it  would  not  slide  on  the  frame,  the  lathe  would  be 
stopped  and  no  work  done  with  it  till  it  was  again  properly 
adjusted.  These  illustrations  might  be  multiplied  indefinitely, 
but  the  reader  will  think  out  many  others  for  himself. 

This  is,  then,  a  distinct  feature  of  the  machine,  that  the  relative 
motions  of  all  parts  are  completely  fixed  and  do  not  depend  in  any 
wajr  upon  the  action  of  external  forces.  Or  perhaps  it  is  better 
to  say  that  whatever  external  forces  are  applied,  the  relative 
paths  of  the  parts  are  unaltered. 

9.  Purpose  of  the  Machine. — There  remains  one  other  matter 
relative  to  the  machine,  and  that  is  its  purpose.     Machines 
are  always  designed  for  the  special  purpose  of  doing  work.     In 
a  steam  engine  energy  is  supplied  to  the  cylinder  by  the  steam 
from  the  boiler,  the  object  of  the  engine  is  to  convert  this  energy 
into  some  useful  form  of  work,  such  as  driving  a  dynamo  or 
pumping  water.     Power  is  delivered  to  the  spindle  of  a  lathe 
through  a  belt,  and  the  lathe  in  turn  uses  this  energy  in  doing 
work  on  a  bar  by  cutting  a  thread.     Energy  is  supplied  to  the 
crank  on  a  windlass,  and  this  energy,  in  turn,  is  taken  up  by  the 
work  done  in  lifting  a  block  of  stone.     Every  machine  is  thus 
designed  for  the  express  purpose  of  doing  work. 

10.  Definition  of  the  Machine. — All  these  points  may  now  be 
summed  up  in  the  form  of  a  definition:  A  machine  consists  of 
resistant  parts,  which  have  a  definitely  known  motion  relative 
to  each  other,  and  are  so  arranged  that  a  given  form  of  available 
energy  may  be  made  to  do  a  desired  form  of  work. 


8  THE  THEORY  OF  MACHINES 

11.  Imperfect  Machines. — Many  machines  approach  a  great 
state  of  perfection,  as  for  example  the  cases  quoted  of  the  steam 
engine  and  the  lathe,  where  all  parts  are  carefully  made  and  the 
motions  are  all  as  close  to  those  desired  as  one  could  make  them. 
But   there   are   many   others,    which   although  commonly  and 
correctly    classed   as  machines,  do  not  come  strictly  under  the 
definition.     Take  the  case  of  the  block  and  tackle  which   will 
be  assumed  as  attached  to  the  ceiling  and  lifting  a  weight. 
In  the  ideal  case  the  pulling  chain  would  always  remain  in  a  given 
position  and  the  weight  should  travel  straight  up  in  a  vertical 
line,  and  in  so  far  as  this  takes  place  the  machine  may  be  con- 
sidered as  serving  its  purpose,  but  if  the  weight  swings,  then 
motion  is  lost  and  the  machine  departs  from  the  ideal  conditions. 
Such  imperfections  are  not  uncommon  in  machines;  the  endlong 
motion  of  a  rotor  of  an  electrical  machine,  the  "flapping"  of  a 
loose  belt  or  chain,  etc.,  are  familiar  to  all  persons  who  have  seen 
machinery  running;  and  even  the  unskilled  observer  knows  that 
conditions  of  this  kind  are  not  good  and  are  to  be  avoided  where 
possible,  and  the  more  these  incorrect  motions  are  avoided,  the 
more  perfect  is  the  machine  and  the  more  nearly  does  it  comply 
with  the  conditions  for  which  it  was  designed. 

DIVISIONS  OF  THE  SUBJECT 

12.  Divisions  of  the  Subject. — It  is  convenient  to  divide  the 
study  of  the  machine  into  four  parts: 

1.  A  study  of  the  motions  occurring  in  the  machine  without 
regard  to  the  forces  acting  externally;  this  study  deals  with  the 
kinematics  of  machinery. 

2.  A  study  of  the  external  forces  and  their  effects  on  the 
parts  of  the  machine  assuming  them  all  to  be  moving  at  uniform 
velocity  or  to  be  in  equilibrium;  the  balancing  forces  may  then 
be  found  by  the  ordinary  methods  of  statics  and  the  problems 
are  those  of  static  equilibrium. 

3.  The  study  of  mechanics  of  machinery  takes  into  account  the 
mass  and  acceleration  of  each  of  the  parts  as  well  as  the  external 
forces. 

4.  The  determination  of  the  proper  sizes  and  shapes  to  be 
given  the  various  parts  so  that  they  may  be  enabled  to  carry 
the  loads  and  transmit  the  forces  imposed  upon  them  from 
without,  as  well  as  from  their  own  mass.     This  is  machine  design, 


THE  NATURE  OF  THE  MACHINE  9 

a  subject  of  such  importance  and  breadth  as  to  demand  an  en- 
tirely separate  treatment,  and  so  only  the  first  three  divisions 
are  dealt  with  in  the  present  treatise. 

KINDS  OF  MOTION 

13.  Plane  Motion. — It  will  be  best  to  begin  on  the  first  division 
of  the  subject,  and  to  discuss  the  methods  adopted  for  obtaining 
definite  forms  of  motion  in  machines.     In  a  study  of  the  steam 
engine,  which  has  already  been  discussed  at  some  length,  it  is 
observed  that  in  each  moving  part  the  path  of  any  point  always 
lies  in  one  plane,  for  example,  the  path  of  a  point  on  the  crankpin 
lies  on  a  plane  normal  to  the  crankshaft,  as  does  also  the  path 
of  any  point  on  the  connecting  rod,  and  also  the  path  of  any 
point  on  the  crosshead.     Since  this  is  the  case,  the  parts  of  an 
engine  mentioned  are  said  to  have  plane  motion,  by  which  state- 
ment is  simply  meant  that  the  path  of  any  point  on  these  parts 
always  lies  in  one  and  the  same  plane.     In  a  completed  steam 
engine  with  slide  valve,  all  parts  have  plane  motion  but  the 
governor  balls,  in  a  lathe  all  parts  usually  have  plane  motion, 
the  same  is  true  of  an  electric  motor  and,  in  fact,  the  vast  majority 
of  the  motions  with  which  one  has  to  deal  in  machines  are  plane 
motions. 

14.  Spheric  Motion. — There  are,  however,  cases  where  different 
motions  occur,  for  example,  there  are  parts  of  machines  where  a 
point  always  remains  at  a  fixed  distance  from  another  fixed  point, 
or  where  the  motion  is  such  that  any  point  will  always  lie  on  the 
surface  of  a  sphere  of  which  the  fixed j*fnt /is  the  center,  as  in 
the  universal  and  ball  and  socket  joints.     Such  motion  is  called 
spheric  motion  and  is  not  nearly  so  common  as  the  plane  motion. 

15.  Screw  Motion. — A  third  class  of  motions  occurs  where  a 
body  has  a  motion  of  rotation  about  an  axis  and  also  a  motion 
of  translation  along  the  axis  at  the  same  time,  the  motion  of 
translation  bearing  a  fixed  ratio  to  the  motion  of  rotation.     This 
motion   is   called   helical   or   screw   motion   and   occurs    quite 
frequently. 

In  the  ordinary  monkey  wrench  the  movable  jaw  has  a,  plane 
motion  relative  to  the  part  held  in  the  hand,  the  plane  motion 
being  one  of  translation  or  sliding,  all  points  on  the  screw  have 
plane  motion  relative  to  the  part  held,  the  motion  being  one  of 
rotation  about  the  axis  of  the  screw,  and  the  screw  has  a  helical 
motion  relative  to  the  movable  jaw,  and  vice  versa. 


10 


THE  THEORY  OF  MACHINES 


PLANE  CONSTRAINED  MOTION 

It  has  been  noticed  already  that  plane  motion  is  frequently 
constrained  by  causing  a  body  to  rotate  about  a  given  axis  or  by 
causing  the  body  to  move  along  a  straight  line  in  a  motion  of 
translation,  the  first  form  of  motion  may  be  called  turning  motion, 
the  latter  form  sliding  motion. 

16.  Turning  Motion. — This  may  be  constrained  in  many  ways 
and  Fig.  1  shows  several  methods,  where  a  shaft  runs  in  a  fixed 
bearing,  this  shaft  carrying  a  pulley  as  shown  in  the  upper  left 


(a) 


Truck  A 


(d) 


B 


FIG.   1. — Forms  of  turning  pairs. 


figure,  while  the  lower  left  figure  shows  a  thrust  bearing  for  the 
propeller  shaft  of  a  boat.  In  the  figure  (a),  there  is  a  pulley  P 
keyed  to  a  straight  shaft  S  which  passes  through  a  bearing  B, 
and  if  the  construction  were  left  in  this  form  it  would  permit 
plane  turning  motion  in  the  pulley  and  shaft,  but  would  not 
constrain  it,  as  the  shaft  might  move  axially  through  B.  If, 
however,  two  collars  C  are  secured  to  the  shaft  by  screws  as 
shown,  then  these  collars  effectually  prevent  the  axial  motion  and 
make  only  pure  turning  possible.  On  the  propeller  shaft  at  (6) 
the  collars  C  are  forged  on  the  shaft,  a  considerable  number  being 


THE  NATURE  OF  THE  MACHINE  11 

used  on  account  of  the  great  force  tending  to  push  the  shaft 
axially.  Thus  in  both  cases  the  relative  turning  motion  is  neces- 
sitated by  the  two  bodies,  the  shaft  with  its  collars  forming  one 
and  the  bearing  the  other,  and  these  together  are  called  a  turning 
pair  for  obvious  reasons,  the  pair  consisting  of  two  elements. 

It  is  evident  that  the  turning  pair  may  be  arranged  by  other 
constructions  such  as  those  shown  on  the  right  in  Fig.  1,  the  form 
used  depending  upon  circumstances.  The  diagram  (c)  shows 
in  outline  the  method  used  in  railroad  cars,  the  bearing  coming 
in  contact  with  the  shaft  only  for  a  small  part  of  the  cir- 
cumference of  the  latter,  the  two  being  held  in  contact  purely 
because  of  the  connection  to  the  car  which  rests  on  top  of  B, 
and  the  collars  C  are  here  of  slightly  different  form.  At  (d) 
is  a  vertical  bearing  which,  in  a  somewhat  better  form  is  often 
used  in  turbines,  but  here  again  it  is  only  possible  to  insure  turn- 
ing motion  provided  the  weight  is  on  the  vertical  shaft  and 
presses  it  into  B.  In  this  case  there  is  only  one  part  correspond- 
ing to  the  collar  C,  which  is  the  part  of  B  below  the  shaft.  At 
(e)  is  a  ball  bearing  used  to  support  a  car  on  top  of  a  truck,  the 
weight  of  the  car  holding  the  balls  in  action. 

17.  Chain  and  Force  Closure. — In  the  cases  (a)  and  (6),  turn- 
ing motion  will  take  place  by  construction,  and  is  said  to  be 
secured  by  chain  closure,  which  will  be  referred  to  later,  while 
in  the  cases  (c),  (d)  and  .(e)  the  motion  is  only  constrained  so  long 
as  the  external  forces  act  in  such  a  way  as  to  press  the  two 
elements  of  the  pair  together,   plane  motion  being  secured  by 
force  closure.     In  cases,  such  as  those  described,  where  force 
closure  is  permissible,  it  forms  the  cheaper  construction,  as  a 
general  rule. 

18.  Sliding  Motion. — The  sliding  pair  also  consists  of  two 
elements,  and  if  a  section  of  these  elements  is  taken  normal  to 
the  direction  of  sliding  the  elements  must  be  non-circular.     As 
in  the  previous  case  the  sliding  pair  in  practice  has  very  many 
forms,  a  few  of  which  are  shown  in  Fig.  2,  (a),  (6),  (c)  and  (d) 
being  forms  in  common  use  for  the  crossheads  of  steam  engines, 
(6)   and   (c)   being  rather  cheaper  in  general  than  the  others. 
At  (e),  (/)  and  (</)  are  shown  forms  which  are  used  in  automobile 
change  gears  and  other  similar  places  where  there  is  little  sliding; 
(e)  consists  of  a  gear  with  a  long  keyway  cut  in  it  while  the  other 
element  has  a  parallel  key,  or  "feather,"  fastened  to  it,  so  that 
the  outer  element  may  slide  along  the  shaft  but  cannot  rotate 


12 


THE  THEORY  OF  MACHINES 


upon  it.  The  construction  of  the  forms  (e)  and  (/)  is  evident. 
The  reader  will  see  very  many  forms  of  this  pair  in  machines  and 
should  study  them  carefully. 


FIG.  2. — Forms  of  sliding  pairs. 

In  the  automobile  engine  and  in  all  the  smaller  gas  and  gasoline 
engines,  the  sliding  pair  is  circular,  because  the  crosshead  is 
omitted  and  the  connecting  rod  is  directly  attached  to  the  piston, 
the  latter  being  circular  and  not  constraining  sliding  motion. 


THE  NATURE  OF  THE  MACHINE 


13 


In  this  case  the  sliding  motion  is  constrained  through  the  con- 
necting rod,  which  on  account  of  the  pairing  at  its  two  ends 
will  not  permit  the  piston  to  rotate.  The  real  sliding  pair,  of 
course,  consists  of  the  cylinder  and  piston,  both  of  which  are 
circular,  and  constrainment  is  by  force  closure. 

In  the  case  of  sliding  pairs  also  it  is  possible  to  have  chain 
closure  where  constraint  is  due  to  the  construction,  as  in  the  cases 
illustrated  in  Fig.  2;  in  these  cases  the  motion  being  one  of 
sliding  irrespective  of  the  directions  of  the  acting  forces.  Fre- 


FIG.  3. — Sliding  pairs. 

quently,  however,  force  closure  is  used  as  in  the  case  (6)  shown  at 
Fig.  3  which  represents  a  planer  table,  the  weight  of  which 
alone  keeps  it  in  place.  Occasionally  through  an  accident  the 
planer  table  may  be  pushed  out  of  place  by  a  pressure  on  the 
side,  but,  of  course,  the  planer  is  not  again  used  until  the  table  is 
replaced,  for  the  reason  that  the  design  is  such  that  the  table 
is  only  to  have  plane  motion,  a  condition  only  possible  if  the 
table  rests  in  the  grooves  in  the  frame.  In  Fig.  3  (a)  the  same 


14 


THE  THEORY  OF  MACHINES 


table  is  constrained  by  chain  closure  and  the  tail  sliding  piece  of 
the  piston  rod  in  Fig.  3  (c)  by  force  closure  as  is  evident. 

19.  Lower  and  Higher  Pairs. — The  two  principal  forms  of 
plane  constrained  motion  are  thus  turning  and  sliding,  these 
motions  being  controlled  by  turning  and  sliding  pairs  respect- 
ively, and  each  pair  consisting  of  two  elements.  Where  contact 
between  the  two  elements  of  a  pair  is  over  a  surface  the  pair  is 
called  a  lower  pair,  and  where  the  contact  is  only  along  a  line 
or  at  a  point,  the  pair  is  called  a  higher  pair.  To  illustrate  this 
the  ordinary  bearing  may  be  taken  as  a  very  common  example 
of  lower  pairing,  whereas  a  roller  bearing  has  line  contact  and 
a  ball  bearing  point  contact  and  are  examples  of  higher  pairing, 
these  illustrations  are  so  familiar  as  to  require  no  drawings.  The 


FIG. 


contact  between  spur  gear  teeth  is  along  a  line  and  therefore  an 
example  of  higher  pairing. 

In  general,  the  lower  pairs  last  longer  than  the  higher,  because 
of  the  greater  surface  exposed  for  wear,  but  the  conditions  of  the 
problem  settle  the  type  of  pairing.  Thus,  lower  pairing  is  used  on 
the  main  shafts  of  large  engines  and  turbines,  but  for  automobiles 
and  bicycles  the  roller  and  ball  bearings  are  common. 


MACHINES,  MECHANISMS,  ETC. 

20.  Formation  of  Machines. — Returning  now  to  the  steam 
engine,  Fig.  4,  its  formation  may  be  further  studied.  The 
valve  gear  and  governor  will  be  omitted  at  present  and  the 
remaining  parts  discussed,  these  consist  of  the  crank,  crankshaft 
and  flywheel,  the  connecting  rod,  the  piston,  piston  rod  and 


THE  NATURE  OF  THE  MACHINE 


15 


crosshead,  and  finally  the  frame  and  cylinder.  Taking  the 
connecting  rod  b  it  is  seen  to  contain  two  turning  elements,  one 
at  either  end,  and  the  real  function  of  the  metal  in  the  rod  is 
to  keep  these  two  elements  parallel  and  at  a  fixed  distance  apart. 
The  crank  and  crankshaft  a  contains  two  turning  elements,  one 
of  which  is  paired  with  one  of  the  elements  on  the  connecting  rod 
6,  and  forms  the  crankpin,  and  the  other  is  paired  with  a  corre- 
sponding element  on  the  frame  d,  forming  the  main  bearing. 
It  is  true  that  the  main  bearing  may  be  made  in  two  parts,  both 
of  which  are  made  on  the  frame,  as  in  center-crank  engines,  or 
one  of  which  may  be  placed  as  an  outboard  bearing,  but  it  will 
readily  be  understood  that  this  division  of  the  bearing  is  only  a 


FIG.  5. — Two-cycle  gasoline 
engine. 


d 
FIG.  6. 


matter  of  practical  convenience,  for  it  is  quite  conceivable  that 
the  bearing  might  be  made  in  one  piece,  and  if  this  piece  were 
long  enough  it  would  serve  the  purpose  perfectly.  Thus  the 
crank  consists  essentially  of  two  turning  elements  properly 
connected. 

Again,  the  frame  d  contains  the  outer  element  of  a  turning 
pair,  of  which  the  inner  element  is  the  crankshaft,  and  it  also 
contains  a  sliding  element  which  is  usually  again  divided  into  two 
parts  for  the  purpose  of  convenience  in  construction,  the  parts 
being  the  crosshead  guides  and  the  cylinder.  But  the  two  parts 
are  not  absolutely  essential,  for  in  the  single-acting  gasoline 


16  THE  THEORY  OF  MACHINES 

engine,  the  guides  are  omitted  and  the  sliding  element  is  entirely 
in  the  cylinder.  Of  course,  the  shape  of  the  element  depends 
upon  the  purpose  to  which  it  is  put;  thus  in  the  case  last  referred 
to  it  is  round. 

Then,  there  is  the  crosshead  c,  with  the  turning  element 
pairing  with  the  connecting  rod  and  the  sliding  element  pairing 
with  the  sliding  element  on  the  frame.  The  sliding  element 
is  usually  in  two  parts  to  suit  those  of  the  frame,  but  it  may  be 
only  in  one  if  so  desired  and  conditions  permit  of  it  (see  Fig.  2) . 

Thus,  the  steam  engine  consists  of  four  parts,  each  part  con- 
taining two  elements  of  a  pair,  in  some  cases  the  elements  being 
for  sliding,  and  in  others  for  turning. 

Again,  on  examining  the  small  gasoline  engine  illustrated  in 
Fig.  5,  it  will  be  seen  that  the  same  method  is  adopted  here  as  in 
the  steam  engine,  but  the  crosshead,  piston  and  piston  rod  are  all 
combined  in  the  single  piston  c.  Further,  in  the  Scotch  yoke, 
Fig.  6,  a  scheme  in  use  for  pumps  of  small  sizes  as  well  as  on  fire 
engines  of  some  makes  and  for  other  purposes,  there  is  the 
crank  a  with  two  turning  elements,  the  piston  and  crosshead  c 
with  two  sliding  elements,  and  the  block  6,  and  the  frame  d, 
each  with  one  turning  element  and  one  sliding  element. 

21.  Links  and  Chains. — The  same  will  be  found  true  in  all 
machines  having  plane  motion;  each  part  contains  at  least  two 
elements,  each  of  which  is  paired  with  corresponding  elements 
on  the  adjacent  parts.  For  convenience  each  of  these  parts 
of  the  machine  is  called  a  Unk,  and  the  series  of  links  so  con- 
nected as  to  give  a  complete  machine  is  called  a  kinematic  chain, 
or  simply  a  chain.  It  must  be  very  carefully  borne  in  mind  that 
if  a  kinematic  chain  is  to  form  part  of  a  machine  or  a  whole 
machine,  then  all  the  links  must  be  so  connected  as  to  have  definite 
relative  motions,  this  being  an  essential  condition  of  the  machine. 

In  Fig.  7  three  cases  are  shown  in  which  each  link  has  two 
turning  elements.  Case  (a)  could  not  form  part  of  a  machine  be- 
cause the  three  links  could  have  no  relative  motion  whatever,  as 
is  evident  by  inspection,  while  at  (6)  it  would  be  quite  impossible 
to  move  any  link  without  the  others  having  corresponding 
changes  of  position,  and  for  a  given  change  in  the  relative  posi- 
tions of  two  of  the  links  a  definite  change  is  produced  in  the 
others.  Looking  next  at  case  (c) ,  it  is  observed  at  once  that  -both 
DC  and  OD  could  be  secured  to  the  ground  and  yet  AB,  BC,  and 
OA  moved,  that  is  a  definite  change  in  AB  produces  no  necessary 


THE  NATURE  OF  THE  MACHINE 


17 


change  in  OD  or  in  CD,  or  one  link  may  move  without  all  the 
others  undergoing  motion  or  relative  change  of  position.  Such 
an  arrangement  could  not  form  part  of  a  machine  because  the 
relative  motions  of  the  parts  are  not  fixed  but  variable  according 
to  conditions.  At  (d)  is  a  chain  which  can  be  used,  because  if 
any  one  link  move  relatively  to  any  other,  all  the  links  move  re- 
latively, or  if  one  link,  say  OD,  is  fastened  to  the  ground  and 
OA  moved,  then  must  all  the  other  links  move. 

22.  Mechanisms. — When  a  chain  is  used  as  a  machine,  usually 
one  of  the  links  acts  as  the  frame  and  is  fixed  to  a  foundation  or 
other  stationary  body.  In  studying  the  motions  of  various 
links  it  is  not  necessary  to  know  the  exact  shape  of  the  links  at  all, 


(a) 


for  the  motion  is  completely  known  if  the  location  and  form  of 
the  pairs  of  elements  is  known.  Thus,  the  actual  link  may  be 
replaced  by  a  straight  bar  which  connects  the  elements  of  the 
link  together,  and  it  will  always  be  assumed  that  this  bar  never 
changes  its  shape  or  length  during  motion.  Thus,  the  chain 
will  be  represented  by  straight  lines  and  a  chain  so  represented 
having  the  relative  motions  of  all  links  completely  constrained 
and  having  one  link  fixed  will  be  called  a  mechanism. 

23.  Simple  and  Compound  Chains. — If  the  links  of  a  chain 
have  only  two  elements  each,  the  chain  is  said  to  be  simple, 
but  if  any  link  has  three  or  more  elements,  as  AB  or  BD,  in  Fig. 
7  (d),  the  chain  is  compound. 

24.  Inversion  of  the  Chain. — Since  in  forming  a  mechanism 
one  link  of  the  chain  is  fixed,  it  would  appear  that  since  any  of 
the  links  may  be  fixed  in  a  given  chain,  it  may  be  possible  to 

2 


18  THE  THEORY  OF  MACHINES 

change  the  nature  of  the  resulting  mechanism  by  fixing  various 
links  successively.  Take  as  an  example  the  mechanism  shown 
at  (1)  Fig.  8,  d  being  the  fixed  link;  here  a  would  describe  a  circle, 
c  would  swing  about  C  and  b  would  have  a  pendulum  motion, 
but  with  a  moving  pivot  B.  If  b  is  fixed  instead  of  d,  a  still 
rotates,  c  swings  about  B  and  d  now  has  the  motion  b  originally 
had,  or  the  mechanism  is  unchanged. 

If  a  is  fixed  then  the  whole  mechanism  may  rotate,  6  and  d 
rotating  about  A  and  0  respectively  as  shown,  and  c  also  rotating, 
the  form  of  the  mechanism  being  thus  changed  to  one  in  which  all 
the  links  rotate.  If,  on  the  other  hand,  c  is  fixed,  then  none  of 
the  links  can  rotate,  but  b  and  d  simply  oscillate  about  B  and  C 
respectively.  The  reader  will  do  well  to  make  a  cardboard  model 
to  illustrate  this  point. 


FIG.  8. — Inversion  of  the  chain. 

The  process  by  which  the  nature  of  the  mechanism  is  altered  by 
changing  the  fixed  link  is  called  inversion  of  the  chain,  and  in 
general,  there  are  as  many  mechanisms  as  there  are  links  in  the 
chain  of  which  it  is  composed,  although  in  the  above  illustra- 
tion there  are  only  three  for  the  four  links. 

25.  Slider-crank  Chain. — This  inversion  of  the  chain  is  very 
well  illustrated  in  case  of  the  chain  used  in  the  steam  engine, 
which  will  be  referred  to  in  future  as  the  slider-crank  chain. 
The  mechanism  is  shown  in  Fig.  9  with  the  crank  a,  connecting 
rod  b  and  piston  c,  the  latter  having  one  sliding  and  one  turning 
element  and  representing  the  reciprocating  masses,  i.e.,  piston, 
piston  rod  and  crosshead.  The  frame  d  is  represented  by  a 
straight  line  and  although  it  is  common,  yet  the  line  of  motion  of  c 
does  not  always  pass  through  0;  however,  as  shown  at  (1),  it 
represents  the  usual  construction  for  the  ordinary  engine.  If 
now,  instead  of  fixing  d,  b  is  fastened  to  the  foundation,  b  being 


THE  NATURE  OF  THE  MACHINE 


19 


the  longer  of  the  two  links  containing  the  two  turning  elements, 
then  a  still  rotates,  c  merely  swings  about  Q  and  d  has  a  swinging 
and  sliding  motion,  and  -if  c  is  a  cylinder  and  a  piston  is  attached 


FIG.  9. — Inversion  of  slider-crank  chain. 

to  d  the  result  is  the  oscillating  engine  as  shown  at  (2)  Fig.  9, 
and  drawn  in  some  detail  in  Fig.  10. 

If  instead  of  fixing  the  long  rod  b  with  the  two  turning  elements, 
the  shorter  rod  a  is  fixed  as  shown  at  (3) ,  then  b  and  d  revolve 


W/////////W//. 
FIG.   10. — Oscillating  engine. 

about  P  and  0  respectively,  and  c  also  revolves  sliding  up  and 
down  on  d.  If  6  is  driven  by  means  of  a  belt  and  pulley  at 
constant  speed,  then  the  angular  velocity  of  d  is  variable  and  the 
device  may  be  used  as  a  quick-return  motion;  in  fact,  it  is  em- 
ployed in  the  Whitworth  quick-return  motion.  The  practical 


20 


THE  THEORY  OF  MACHINES 


form  is  also  shown,  Fig.  11,  and  the  relation  between  the  mechan- 
ism and  the  actual  machine  will  be  readily  discovered  with  the 
help  of  the  same  letters. 


FIG.   11. — Whitworth  quick-return  motion. 

In  the  Whitworth  quick-return  motion,  Fig.  11,  the  pinion 
is  driven  by  belt  and  meshes  with  the  gear  6.  The  gear  rotates 
on  a  large  bearing  E  attached  to  the  frame  a  of  the  machine,  and 
through  the  bearing  E  is  a  pin  F,  to  one  side  of  the  center  of  E, 


FIG.  12. — Gnome  aeroplane  motor. 

carrying  the  piece  d,  the  latter  being  driven  from  6  by  a  pin  c 
working  in  a  slot  in  d.  The  arm  A  is  attached  to  a  tool  holder 
at  B. 


THE  NATURE  OF  THE  MACHINE 


21 


The  Gnome  motor  used  on  aeroplanes  is  also  an  example  of 
this  same  inversion.  It  is  shown  in  Fig.  12  and  the  cylinder 
shown  at  the  top  with  its  rod  and  piston  form  the  same  mechanism 
as  the  Whitworth  quick-return  motion,  a  being  the  link  between 
the  shaft  and  lower  connecting-rod  centers.  Study  the  mechanism 
used  with  the  other  cylinders. 

The  fourth  inversion  found  by  fixing  c  is  rarely  used  though  it  is 
found  occasionally.  It  is  shown  at  (4)  Fig.  9. 

There  are  thus  four  inversions  of  this  chain  and  it  might  be 
further  changed  slightly  by  placing  Q  to  one  side  of  the  link  d 


FIG.  13. — Shaper  mechanism. 

so  that  the/line  of  motion  of  Q,  Fig.  9  (1),  passses  above  0,  giving 
the  scheme  used  in  operating  the  sleeves  in  some  forms  of  gasoline 
engines,  etc. 

A  somewhat  different  modification  of  the  slider-crank  chain  is 
shown  at  Fig.  13  a  device  also  used  -as  a  quick-return  motion  in 
shapers  and  other  machines.  On  comparing  it  with  the  Whit- 
worth  motion  shown  at  Fig.  11,  and  the  engine  shown  at  Fig.  10, 
it  is  seen  that  the  mechanism  of  Fig.  9  may  be  somewhat  altered 
by  varying  the  proportions  of  the  links.  The  mechanism  illus- 
trated at  Fig.  13  should  be  clear  without  further  explanation. 
D  is  the  driving  pinion  working  in  with  the  large  gear  b,  the  tool 


22 


THE  THEORY  OF  MACHINES 


is  attached  to  B  which  is  driven  from  c  by  the  link  A.  It  is 
readily  seen  that  B  moves  faster  in  one  direction  than  the  other. 
Further,  an  arrangement  is  made  for  varying  the  stroke  of  B 
at  pleasure  by  moving  the  center  of  c  closer  to,  or  further  from, 
that  of  6. 

26.  Double  Slider-crank  Chain. — A  further  illustration  of  a 
chain  which  goes  through  many  inversions  in  practice  is  given 
in  Fig.  14  and  contains  two  links,  b  and  d,  with  one  sliding  and 
one  turning  element  each,  also  one  link  a 
with  two  turning  elements  and  one  c  with 
two  sliding  elements.  When  the  link  d 
is  fixed,  c  has  a  reciprocating  motion  and 
such  a  setting  is  frequently  used  for  small 
pumps  driven  by  belt  through  the  crank 
a  (Fig.  14),  c  being  the  plunger.  A  detail 
of  this  has  already  been  given  in  Fig.  6. 

With  a  fixed  the  device  becomes  Oldham's  coupling  which  is 
used  to  connect  two  parallel  shafts  nearly  in  line,  Fig.  15.  In 
the  figure  b  and  d  are  two  shafts  which  are  parallel  and  rotate 
about  fixed  axes.  Keyed  to  each  shaft  is  a  half  coupling  with  a 
slot  running  across  the  center  of  its  face  and  between  these  half 
couplings  is  a  peice  c  with  two  keys  at  right  angles  to  each  other, 
one  on  each  side,  fitting  in  grooves  in  b  and  d.  As  b  and  d 


FIG.  14. 


FIG.   15. — Oldham's  coupling. 

revolve,  c  works  sideways  and  vertically,  both  shafts  always  turn- 
ing at  the  same  speed.  Points  on  c  describe  ellipses  and  a  modi- 
fication of  the  device  has  been  used  on  elliptical  chucks  and  on 
instruments  for  drawing  ellipses. 


QUESTIONS  ON  CHAPTER  I 

1.  Define  the  term  machine  and  show  that  a  gas  engine,  a  stone  crusher 
and  a  planer  are  machines.     Is  a  plough  or  a  hay  rake  or  hay  fork  a  ma- 
chine?    Why? 

2.  What  are  the  methods  of  constrainment  employed  in  the  following: 


THE  NATURE  OF  THE  MACHINE  23 

Line  shafting,  loose  pulley,  sprocket  chain,  engine  crankshaft,  lathe  spindle, 
eccentric  sheave,  automobile  clutch,  change  gear,  belt.  Which  are  by  force 
and  which  by  chain  closure? 

3.  Make  a  classification  of  the  following  with  regard  to  constrainment 
and  the  form  of  closure:  Gas-engine  piston,  lathe  carriage,  milling-machine 
head,  ordinary  D-slide  valve,  locomotive  crosshead,  valve  rod,  locomotive 
link.     Give  a  sketch  to  illustrate  each.     Why  would  force  closure  not  do 
for  a  connecting  rod? 

4.  What  form  of  pairing  is  used  in  the  cases  given  in  the  above  two  ques- 
tions?    Is  lower  or  higher  pairing  used  in  the  following,  and  what  is  the 
type  of  contact:   Roller  bearing,  ball  bearing,  vertical-step  bearing,  cam  and 
roller  in  sewing  machine,  gear  teeth,  piston? 

6.  Define  plane,  helical  and  spherical  motion.     What  form  is  used  in  the 
parts  above  mentioned,  and  in  a  pair  of  bevel  gears? 

6.  In  helical  motion  if  the  pitch  of  the  helix  is  zero,  what  form  of  motion 
results;  also  what  form  for  infinite  pitch? 

7.  What  is  the  resulting  form  of  motion  if  the  radius  for  a  spherical 
motion  becomes  infinitely  great? 

8.  Show  that  all  the  motions  in  an  ordinary  engine  but  that  of  the  gover- 
nor balls  are  plane.     What  form  of  motion  do  the  latter  have? 

9.  Define  and  illustrate  the  following  terms :  Element,  lower  pair,  higher 
pair,  link,  chain,  mechanism  and  compound  chain. 

10.  List  the  links  and  their  elements  and  give  the  form  of  motion  and 
method  of  constrainment  in  the  parts  of  a  locomotive  side  rod,  beam  engine, 
stone  crusher  (Fig.  95)  and  shear  (Fig.  94). 

11.  Explain  and  illustrate  the  inversion  of  the  chain.     Show  that  the 
epicyclic  gear  train  is  an  inversion  of  the  ordinary  train. 


CHAPTER  II 
MOTION  IN  MACHINES 

27.  Plane  Motion. — It  is  now  desirable  to  study  briefly  certain 
of  the  characteristics  of  plane  motion,  a  term  which  may  be 
defined  by  stating  that  a  body  has  plane  motion  when  it  moves  in 
such  a  way  that  any  given  point  in  it  always  remains  in  one  and 
the  same  plane,  and  further,  that  the  planes  of  motion  of  all 
points  in  the  body  are  parallel.  Thus,  if  any  body  has  plane 
motion  relative  to  the  paper,  then  any  point  in  the  body  must 
remain  in  a  plane  parallel  to  the  plane  of  the  paper  during  the 
motion  of  the  body. 

A  little  consideration  will  show  that  in  the  case  of  plane  motion 
the  location  of  a  body  is  known  when  the  location  of  any  line  in 
the  body  is  known,  provided  this  line  lies  in  a  plane  parallel  to 
the  plane  of  motion  or  else  in  the  plane  of  motion  itself.  The 
explanation  is,  that  since  all  points  in  the  body  have  plane  motion, 
then  the  projection  of  the  body  on  the  plane  is  always  the  same 
for  all  positions  and  hence  the  line  in  it  simply  locates  the  body. 
For  example,  if  a  chair  were  pushed  about  upon  the  floor  and  had 
points  marked  R  and  L  upon  the  bottoms  of  two  of  the  legs, 
then  the  location  of  the  chair  is  always  known  if  the  positions 
of  R  and  L,  that  is,  of  the  (imaginary)  line  RL  is  known.  If, 
however,  the  chair  were  free  to  go  up  and  down  from  the  floor 
it  would  be  necessary  to  know  the  position  of  the  projection  of 
RL  on  the  floor  and  also  the  height  of  the  line  above  the  floor 
at  any  instant.  Further,  if  it  were  possible  for  the  chair  to  be 
tilted  backward  about  the  (imaginary)  line  RL,  the  position 
of  the  latter  would  tell  very  little  about  the  position  of  the  chair, 
as  the  tips  of  its  legs  might  be  kept  stationary  while  tilting  the 
chair  back  and  forth,  the  position  of  RL  being  the  same  for  various 
angular  positions  of  the  chair. 

If  the  case  where  a  body  has  not  plane  motion  is  considered, 
then  the  line  will  tell  very  little  about  the  position  of  the  body. 
In  the  case  of  an  airship,  for  example,  the  ship  may  stand  at 
various  angles  about  a  given  line,  say  the  axis  of  a  pair  of  the 

24 


MOTION  IN  MACHINES  25 

wheels,  the  ship  dipping  downward  or  rising  at  the  will  of  the 
operator. 

28.  Motion  Determined  by  that  of  a  Line. — Since  the  location 
of  a  body  having  plane  motion  is  known  when  the  location  of 
any  line  in  the  body  is  known,  then  the  motion  of  the  body  will 
be  completely  known,  if  the  motion  of  any  line  in  the  body  is 
known.     Thus  let  C,  Fig.   16,  represent  the  projection  on  the 
plane  of  the  paper  of  any  body  having  plane  motion,  AB  being 
any  line  in  this  body,  and  let  AB  be  assumed  to  be  in  the  plane  of 
the  paper,  which  is  used  as  the  plane  of  reference.     Suppose  now 
it  is  known  that  while  C  moves  to  C",  the  points  A  and  B  move  over 
the  paths  A  A'  and  BB',  then  the  motion  of  C  during  the  change 
is  completely  known.     Thus  at  some  intermediate  position  the 
line  is  at  AiBi  and  the  figure  of  C  can  at  once  be  drawn  about 
this  line,  and  this  locates  the  posi- 
tion of  the  body  corresponding  to 

the  location  A i#i  of  the  line  AB. 

It  will  therefore  follow  that  the 
\motion  of  a  body  is  completely 
V  known  provided  only  that  the 
/  motion  of  any  line  in  the  body  is  FlG  16 

v  known.     This    proposition    is    of 

much  importance  and  should  be  carefully  studied  and  understood. 

29.  Relative  Motion. — It   will  be  necessary  at  this  point  to 
grasp  some  idea  of  the  meaning  of  relative  motion.     We  have 
practically  no  idea  of  any  other  kind  of  motion  than  that  referred 
to  some  other  body  which  moves  in  space,  the  moon  is  said  to 
move  simply  because  it  changes  its  position  as  seen  from  the  earth, 
or  a  train  is  said  to  move  as  it  passes  people  standing  on  a  rail- 
road crossing.     Again,  one  sees  passengers  in  a  railroad  car  as 
the  train  moves  out  and  says  they  are  moving,  while  each  pas- 
senger in  turn  looks  at  other  passengers  sitting  in  the  same  car 
and  says  the  latter  are  still.     Again,  a  brakeman  may  walk 
backward  on  a  flat  car  at  exactly  the  same  rate  as  the  car  goes 
forward,  and  a  person  on  the  ground  who  could  just  see  his  head 
would  say  he  was  stationary,  while  the  engine  driver  would  say 
he  was  moving  at  several  miles  per  hour.     If  one  stood  on  shore 
and  saw  a  ship  go  out  one  would  say  that  the  funnel  was  moving, 
and  yet  a  person  on  the  ship  would  say  that  it  was  stationary. 

These  conflicting  statements,  which  are,  however,  very  com- 
mon, would  lead  to  endless  confusion  unless  the  essential  differ- 


26  THE  THEORY  OF  MACHINES 

ences  in  the  various  cases  were  grasped,  and  it  will  be  seen  that 
the  real  difference  of  view  results  from  the  fact  that  different 
persons  have  entirely  different  standards  of  comparison.  Stand- 
ing on  the  ground  the  standard  of  rest  is  the  earth,  and  anything 
that  moves  relative  to  it  is  said  to  be  moving.  The  man  on  the 
flat  car  would  be  described  as  stationary  because  he  does  not 
move  with  regard  to  the  chosen  standard — the  earth,  but  the 
engine  driver  would  be  thinking  of  the  train,  and  he  would  say 
the  man  moved  because  he  moved  relative  to  his  standard — the 
train.  It  is  easy  to  multiply  these  illustrations  indefinitely, 
but  they  would  always  lead  to  the  same  result,  that  whether  a 
body  moves  or  remains  at  rest  depends  altogether  upon  the 
standard  of  comparison,  and  it  is  usual  to  say  that  a  body  is  at 
rest  when  it  has  the  same  motion  as  the  body  on  which  the 
observer  stands,  and  that  it  is  in  motion  when  its  motion  is 
different  to  that  of  the  body  on  which  the  observer  stands. 
On  a  railroad  train  one  speaks  of  the  poles  flying  past,  whereas  a 
man  on  the  ground  says  they  are  fixed. 

30.  Absolute  and  Relative  Motion. — When  the  standard  which 
is  used  is  the  earth  it  is  usual  to  speak  of  the  motions  of  other 
bodies    as    absolute    (although   this  is  incorrect,  for  the  earth 
itself  moves)  and  when  any  standard  which  moves  on  the  earth 
is  used,  the  motions  of  the  other  bodies  are  said  to  be  relative. 
Thus  the  absolute  motion  of  a  body  is  its  motion  with  regard 
to  the  earth,  and  the  relative  motion  is  the  motion  as  compared 
with  another  body  which  is  itself  moving  on  the  earth.     Unless 
these  ideas  are  fully  appreciated  the  reader  will  undoubtedly  meet 
with  much  difficulty  with  what  follows,  for  the  notion  of  relative 
motion  is  troublesome. 

In  this  connection  it  should  be  pointed  put  that  a  body  secured 
to  the  earth  may  have  motion  relative  to  another  body  which  is 
not  so  secured.  Thus  when  a  ship  is  coming  into  port  the  dock 
appears  to  move  toward  the  passengers,  but  to  the  person  on 
shore  the  ship  appears  to  come  toward  the  shore,  thus  the  motion 
of  the  ship  relative  to  the  dock  is  equal  and  opposite  to  the  motion 
of  the  dock  relative  to  the  ship. 

31.  Propositions  Regarding  Relative  Motion. — Certain  proposi- 
tions will  now  be  self-evident,  the  first  being  that  if  two  bodies 
have  no  relative  motion  they  have  the  same  motion  relative  to 
every  other  body.     Thus,  two  passengers  sitting  in  a  train  have 
no  relative  motion,  or  do  not  change  their  positions  relative  to 


MOTION  IN  MACHINES  27 

one  another,  and  hence  they  have  the  same  motion  or  cnange  of 
position  relative  to  the  earth,  or  to  another  train  or  to  any  other 
body:  the  converse  of  this  proposition  is  also  true,  or  two  bodies 
which  have  the  same  change  of  position  relative  to  other  bodies 
have  no  relative  motion. 

32.  Another  very  important  proposition  may  be  stated  as 
follows:  The  relative  motions  of  two  bodies  are  not  affected  by 
any  motion  which  they  have  in  common.  Thus  the  motion  of 
the  connecting  rod  of  an  engine  relative  to  the  frame  is  the 
same  whether  the  engine  is  a  stationary  one,  or  is  on  a  steamboat 
or  locomotive,  simply  because  in  the  latter  cases  the  motion  of 
the  locomotive  or  ship  is  common  to  the  connecting  rod  and 
frame  and  does  not  affect  their  relative  motions. 

The  latter  proposition  leads  to  the  statement  that  if  it  be 
desired  to  study  the  relative  motions  in  any  machine  it  will  not 
produce  any  change  upon  them  to  add  the  same  motion  to  all 
parts.  For  example,  if  a  bicycle  were  moving  along  a  road  it 
would  be  found  almost  impossible  to  study  the  relative  motions 
of  the  various  parts,  but  it  is  known  that  if  to  all  parts  a  motion 
be  added  sufficient  to  bring  the  frame  to  rest  it  will  not  in  any 
way  affect  the  relative  motions  of  the  parts  of  the  bicycle. 
Or  if  it  be  desired  to  study  the  motions  in  a  locomotive  engine, 
then  to  all  parts  a  common  motion  is  added  which  will  bring 
one  part,  usually  the  frame,  to  rest  relatively  to  the  observer,  or  to 
the  observer  and  to  all  parts  of  the  machine  such  a  motion  is 
added  as  to  bring  him  to  rest  relative  to  them,  in  fact,  he  stands 
upon  the  engine,  having  added  to  himself  the  motion  which  all 
parts  of  the  engine  have  in  common.  So  that,  whenever  it  is 
found  necessary  to  study  the  motions  of  machines  all  parts  of 
which  are  moving,  it  will  always  be  found  convenient  to  add  to  the 
observer  the  common  motion  of  all  the  links,  which  will  bring 
one  of  them  to  rest,  relative  to  him. 

To  give  a  further  illustration,  let  two  gear  wheels  a  and  6  run 
together  and  turn  in  opposite  sense  about  fixed  axes.  Let  a  run 
at  +  50  revolutions  per  minute,  and  b  at  —  80  revolutions  per 
minute;  it  is  required  to  study  the  motion  of  b  relative  to  a. 
To  do  this  add  to  each  such  a  motion  as  to  bring  a  to  rest,  that 
is,  —  50  revolutions  per  minute,  the  result  being  that  a  turns 
+  50  —  50  =  0,  while  b  turns  —  80  —  50  =  —  130  revolutions 
per  minute  or  b  turns  relative  to  a  at  a  speed  of  130  revolutions 
per  minute  and  in  opposite  sense  to  a.  Here  there  has  simply 


28  THE  THEORY  OF  MACHINES 

been  added  to  each  wheel  the  same  motion,  which  does  not  affect 
their  relative  motions  but  has  the  effect  of  bringing  one  of  the 
wheels  to  rest.  To  find  the  motion  of  a  relative  to  6,  bring  b 
to  rest  by  adding  +  80  revolutions  per  minute,  so  that  a  goes 
+  50  +  80  =  130  revolutions  per  minute,  or  the  motion  of  b 
relative  to  a  is  equal  and  opposite  to  that  of  a  relative  to  b. 

33.  The  Instantaneous  or  Virtual  Center. — It  has  already  been 
pointed  out  in  Sec.  27  that  when  a  body  has  plane  motion,  the 
motion  of  the  body  is  completely  known  provided  the  motion 
of  any  line  in  the  body  in  the  plane  of  motion  is  known,  that  is, 
provided  the  motions  or  paths  of  any  two  points  in  the  body  are 
known.  Now  let  c,  Fig.  17,  represent  any  body  moving  in  the 
plane  of  the  paper  at  any  instant,  the  line  AB  being  also  in  the 
plane  of  the  paper,  and  let  FA /  and  BE  repre- 
sent short  lengths  of  the  paths  of  A  and  B 
respectively  at  this  instant.  The  direction  of 
motion  of  A  is  tangent  to  the  path  FA  at  A, 
and  that  of  B  is  tangent  to  the  path  BE  at 
B,  the  paths  of  A  and  B  giving  at  once  the 
direction  of  the  motions  of  these  points  at  the 
instant.  Through  A  draw  a  normal  AO  to 
the  direction  of  motion  of  A,  then,  if  a  pin  is 
stuck  through  any  point  on  the  line  AO  into 
the  plane  of  reference  and  c  is  turned  very 
slightly  about  the  pin  it  will  give  to  A  the 
direction  of  motion  it  actually  has  at  the  instant.  The  same 
argument  applies  to  BO  a  normal  to  the  path  at  B,  and  hence 
to  the  point  0  where  AO  and  BO  intersect,  that  is,  if  a  pin  is  put 
through  the  point  0  in  the  body  c  and  into  the  plane  of  reference, 
where  the  body  is  in  the  position  shown,  the  actual  motion  of 
the  body  is  the  same  as  if  it  rotated  for  an  instant  about  this 
pin.  0  is  called  the  instantaneous  or  virtual  center,  because  it 
is  the  point  in  the  body  c  about  which  the  latter  is  virtually 
turning,  with  regard  to  the  paper,  at  the  instant. 

In  going  over  this  discussion  it  will  appear  that  0  may  be  found 
provided  only  the  directions  of  motion  of  A  and  B  are  known  at 
the  instant.  The  only  purpose  for  which  the  paths  of  these 
points  have  been  used  was  to  get  the  directions  in  which  A  and  B 
were  moving  at  the  instant,  and  the  actual  path  is  unimportant 
in  so  far  as  the  finding  of  0  is  concerned.  It  will  further  appear 
that  the  point  0  will,  in  general,  change  for  each  new  position 


MOTION  IN  MACHINES  29 

of  the  body,  because  the  directions  of  motion  of  A  and  B  will 
be  such  as  to  change  the  location  of  0.  Should  it  happen, 
however,  that  A  and  B  moved  in  parallel  straight  lines,  0  would 
be  at  infinity  or  the  body  c  would  have  a  motion  of  translation; 
on  the  other  hand,  if  the  points  A  and  B  moved  around  concen- 
tric circles,  0  would  be  fixed  in  position,  being  the  common  center 
of  the  two  circles,  and  c  would  simply  rotate  about  the  fixed 
point  0. 

34.  Directions   of   Motion   of   Various   Points. — The   virtual 
center  so  found  gives  much  information  about  the  motion  of  the 
body  at  the  given  instant.     In  the  first  place  it  shows  that  the 
direction  of  motion  of  G,  with  respect  to  the  paper,  which  has 
been  selected  as  the  reference  plane,  is  perpendicular  at  OG  and 
that  of  H  is  perpendicular  to  OH,  since  the  direction  of  motion 
of  any  point  in  a  rotating  body  is  perpendi&ilar  to  the  radius 
to  the  point;  thus,  when  the  virtual  center  is  known,  the  direc- 
tion of  motion  of  every  point  in  the*  body  is  known.     It  is  not 
possible  to  put  down  at  random  to  direction  of  motion  of  G  as 
well  as  those  of  A  and  B  because  that  of  G  is  fixed  when  those  of  A 
and  B  are  given;  the  virtual  center  does  not,  however,  give  the 
path  of  G  but  only  its  direction. 

35.  Linear  Velocities. — In  the  next  place  the  virtual  center 
gives  the  relative  linear  velocities  of  all  points  in  c  at  the  instant. 
Let  the  body  c  be  turning  at  the  rate  of  n  revolutions  per  minute, 
corresponding  to  co  radians  per  second,  the  relation  being  co  = 

-gQ-  •     At  the  instant  the  velocity  v  of  a  point  situated  r  ft.  from 

0  will  evidently  be  v  =     ^     =  rco   ft.    per   second,  and,  since 

co  is  the  same  for  the  whole  body,  the  linear  velocity  of  any  point 
is  proportional  to  its  distance  from  the  center  0. 

Thus  if  VA,  VB,  VQ  be  used  to  denote  the  velocities  of  the  points 
A,  B  and  G  respectively  then  it  follows  that  VA  =  OA-co,  VB  = 
OB-u  and  VG  =  OG-u,  and  it  will  also  be  clear  that  even  though 
oj  is  unknown  the  relations  between  the  three  velocities*  are 
known  and  also  the  sense  of  the  motion. 

36.  Information  Given  by  Virtual  Centers. — The  virtual  center 
for  a  body  may,  therefore,  be  found,  provided  only  that  the 
directions  (not  necessarily  the  paths)  of  motion  of  two  points 
in  it  are  known,  and  having  found  this  center  the  directions  of" 
motion  of  all  points  in  the  body  are  known,  and  their  relative 


30  THE  THEORY  OF  MACHINES 

velocities;  and  also  the  actual  velocities  in  magnitude,  sense  and 
direction  will  be  known  if  the  angular  velocity  is  known.  (This 
should  be  compared  with  the  phorograph  discussed  in  a  later 
chapter.)  It  is  to  be  further  noted  that  the  virtual  center  0 
is  a  double  point ;  it  is  a  point  in  the  paper  and  also  in  c,  and  the 
motion  of  any  point  in  c  with  regard  to  the  paper  being  perpen- 
dicular to  the  radius  from  0  to  that  point  so  also  the  motion  of 
any  point  in  the  paper  with  regard  to  c  is  perpendicular  to  the 
line  joining  this  point  to  0. 

Another  point  is  to  be  noticed,  that  if  the  various  virtual  centers 
0  are  known,  then  at  once  the  relative  motion  of  c  to  the  paper 
is  known.  Thus  the  virtual  center  of  one  body  with  regard  to 
another  gives  always  the  motion  of  the  one  body  with  regard  to 
the  other. 

37.  The  Permanent  Center. — It  has  already  been  pointed  out 
that  the  instantaneous  or  virtual  center  is  the  center  for  rota- 
tion of  any  one  body  with  regard  to  another  at  a  given  instant, 
and  that  the  location  of  this  center  is  changing  from  one  instant 
to  the  next.     There  are,  however,  very  many  cases  where  one 
body  is  joined  to  another  by  means  of  a  regular  bearing,  as  in 
the  case  of  the  crankshaft  of  an  engine  and  the  frame,  or  a  wagon 
wheel  and  the  body  of  the  axle,  or  the  connecting  rod  and  crank- 
pin  of  an  engine.     A  little  reflection  will  show  that  in  each  of  these 
cases  the  one  body  is  always  turning  with  regard  to  the  other,  and 
that  the  center  or  axis  of  revolution  has  a  fixed  position  with 
regard  to  each  of  the  bodies  concerned,  thus  in  these  cases  the 
virtual  center  remains  relatively  fixed  and  may  be  termed  the 
permanent  center. 

38.  The  Fixed  Center. — The  permanent  center  must  not  be 
confused  with  the  fixed  center,  which  term  would  be  applied  to  a 
center  fixed  in  place  on  the  earth,  but  is  intended  to  include  only 
the  case  where  the  virtual  center  for  the  rotation  of  one  body  with 
regard  to  another  is  a  point  which  remains  at  the  same  place  in 
each  body  and  does  not  change  from  one  instant  to  another. 
The  center  between  the  connecting  rod  and  crank  and  between 
the  crankshaft  and  frame  are  both  permanent,  the  latter  being 
also  fixed  usually. 

39.  Theorem   of  the   Three   Centers. — Before   applying   the 
virtual  centers  in  the  solution  of  problems  of  various  kinds,  a 
very  important  property  connected  with  them  will  be  proved. 
Let  a,  b  and  c,  Fig.  18,  represent  three  bodies  all  of  which  have 


MOTION  IN  MACHINES  31 

plane  motion  of  any  nature  whatever,  and  which  motion  is  for 
the  time  being  unknown.  Now,  generally  a  has  motion  relative 
to  b,  and  b  has  motion  relative  to  c,  and  similarly  c  with  regard 
to  a,  in  brief  all  three  bodies  move  in  different  ways,  hence  from 
what  has  been  said  in  Sec.  33,  there  is  a  virtual  center  of  a  ~  b1 
which  may  be  called  ab,  and  this  is  of  course  also  the  center 
of  b  ~  a.  Further,  there  is  a  virtual  center  of  6  ~  c,  that  is  be, 
and  also  a  center  of  c  <~  a,  which  is  ca,  thus  for  the  three  bodies 
there  are  three  virtual  centers.  Now  it  will  be  assumed  that 
enough  information  has  been  given  about  the  motions  of  a,  b 
and  c  to  determine  ab  and  ac  only,  and  it  is  required  to  find  be. 

Since  be  is  a  point  common  to  both  bodies  b  and  c,  let  it  be 
supposed  to  lie  at  P'f  then  P'  is  a  point  in  both  the  bodies  b  and  c. 
As  a  point  in  b  its  motion 
with  regard  to  a  will  be  nor- 
mal to  _jPl^jO&4_  that  is,    in 
the    direction    P'A,    because 
the  motion  of  a  point  in  one 
body  with  regard  to  another 
body   is   normal   to   the   line 
joining  this  point  to  the  vir- 
tual center  for  the  two  bodies  FIG.  18. 
(Sec.   34).     As  a  point  in  c, 

the  motion  of  Pf  ~  a  is  normal  to  Pf  -  ac  or  in  the  direction  P'B, 
j3o  that  P'  has  two  different  motions  with  regard  to  a  at  the 
same  time,  which  is  impossible  or  P'  cannot  be  the  virtual 
center  of  6  ^  c.  Since,  however,  this  is  not  the  point,  it  shows 
at  once  that  the  point  be  is  located  somewhere  along  the  line 
ab  -  ca,  or  say  at  P,  because  it  is  only  such  a  point  as  P  which 
has  the  same  motion  with  regard  to  a  whether  considered  as  a 
point  in  b  or  in  c;  thus  the  center  be  must  lie  on  the  same  straight 
line  as  the  centers  ab  and  ac.  It  is  not  possible  to  find  the  ex- 
act position  of  be,  however,  without  further  information,  all 
that  is  known  is  the  line  on  which  it  lies. 

This  proposition  may  be  thus  stated:  If  in  any  mechanism 
there  are  any  three  links  a,  /,  g,  all  having  plane  motion,  then  for 
the  three  links  there  are  three  virtual  centers  of,  fg  and  ag,  and 
these  three  centers  must  all  He  on  one  straight  line. 

Two  of  the  centers  may  be  permanent  but  not  the  third;  in 

1  The  sign  ~  means  "with  regard  to." 


32  THE  THEORY  OF  MACHINES 

the  steam  engine  taking  the  crank  a,  the  connecting  rod  b  and  the 
frame  d,  the  centers  ab  and  ad  are  permanent,  but  bd  is  not. 

40.  The  Locating  of  the  Virtual  Centers.— The  chapter  will 
be  concluded  by  finding  the  virtual  centers  in  a  few  mechanisms 
simply  to  illustrate  the  method,  the  application  being  given  in 
the  next  chapter.  As  an  example,  consider  the  chain  with  four 
turning  pairs,  which  is  first  taken  on  account  of  its  simplicity. 
It  is  shown  in  Fig.  19,  and  consists  of  four  links,  a,  b,  c  and  d, 
of  different  lengths,  d  being  fixed,  and  by  inspection  the  four 
permanent  centers  ab,  be,  cd  and  ad,  at  the  four  corners  of  the 
chain,  are  at  once  located.  It  is  also  seen  that  there  are  six 
possible  centers  in  the  mechanism,  viz.,  ab,  be,  cd,  da,  bd  and  ac, 


FIG.  19. 

these  being  all  the  possible  combinations  of  the  links  in  the 
chain  when  taken  in  pairs,  and  of  these  six,  the  four  permanent 
ones  are  found  already,  and  only  two  others,  ac  and  bd,  remain. 
There  are  two  methods  of  finding  them,  the  first  of  which  is  the 
most  instructive,  and  will  be  given  first  for  that  reason. 

By  the  principle  of  the  virtual  center  bd  may  be  found  if  the  direc- 
tions of  motion  of  any  two  points  in  b  ~  d  are  known.  On  exam- 
ining ab  remember  that  it  is  a  point  in  a  and  also  in  6;  as  a  point 
in  a  it  moves  with  regard  to  d  about  the  center  ad  and  thus  in  a 
direction  normal  to  ad  -  ab  or  to  a  itself.  And  as  a  point  in  6 
it  must  have  the  same  motion  with  regard  to  ~d  as  it  has  when 
considered  as  a  point  in  a;  that  is,  the  motion  of  ab  in  b  with  re- 
gard to  d  is  in  the  direction  perpendicular  to  a.  Hence,  from 
Sec.  33,  the  virtual  center  will  lie  on  the  line  through  ab  in  the 
direction  of  a,  that  is,  in  a  produced.  Again  be  is  a  point  in  b 
and  c,  and  as  a  point  in  c  it  moves  with  regard  to  d  in  a  direction 


bd 


MOTION  IN  MACHINES  33 

perpendicular  to  cd  -  be,  or  in  the  direction  be  -  F,  and  this  must 
also  be  the  direction  of  be  as  a  point  in  b  ~  d,  so  that  the  virtual 
center  of  b  ~  d  must  also  lie  in  the  line  through  be  normal  to 
be  -  F,  or  in  c  produced.  Hence,  bd  is  at  the  intersection  of  a 
and  c  produced. 

This  could  have  been  solved  by  the  theorem  of  the  three 
centers,  for  there  will  be  three  centers,  ad,  ab  and  bd,  for  the  three 
bodies  a,  b  and  d,  and  these  must  lie  in  one  straight  line,  and  as 
both  ad  and  ab  are  known,  this  gives  the  line  on  which  bd  lies. 
Similarly,  by  considering  the  three  bodies,  b,  c  and  d}  and  know- 
ing the  centers  be  and  cd,  another  line  on  which  bd  lies  is  iso- 
lated, and  hence  bd  is  readily  found.  To  find  the  center  ac  it 
is  possible  to  proceed  in  either  of  the  ways  already  explained, 
and  thus  find  ac  at  the  intersection  of  the  lines  b  and  d  produced. 

41.  Sliding  Pairs. — One  other  example  may  be  solved,  and  in 
order  to  include  a  sliding  pair  consider  the  case  shown  in  Fig.  20, 
in  which  a  is  the  crank,  b  the  connect- 
ing rod,   c  the  crosshead,  piston,   etc.,  cdo° 
and    d    the    fixed    frame.     As    before 
there  are  six  centers  ad,  ab,  be,  cd,  ac, 
bd,   of  which  ad,  ab,  and  be  are  perma- 
nent and  found  by  inspection. 

To   find   the   center  cd  it  is  noticed 
that  c  slides  in  a  horizontal  direction    ad 
with  regard  to  d,  that  is,  c  has  a  motion  FIG.  20. 

of  translation  in  a  horizontal  straight 

line,  or,  what  is  the  same  thing,  it  moves  in  a  circle  of  infinite 
radius,  and  the  center  of  this  circle  must,  as  before,  lie  in  a  line 
normal  to  the  direction  of  motion  of  c  ~  d.  Hence  cd  lies  in 
a  vertical  line  through  be  or  through  any  other  point  in  the 
mechanism  such  as  ad,  and  at  an  infinite  distance  away.  The 
figure  shows  cd  above  the  mechanism,  but  it  might  be  below 
just  as  well.  . 

Having  found  (cd,j  the  other  centers  ac  and  bd  may  be  found  by 
the  theorem  of  the  three  centers.  Thus  bd  lies  on  be  -  cd  and  on 
ad  -  ab  and  'is  therefore  at  their  intersection,  and  similarly  -ac 
lies  at  the  intersection  of  be  -  ab  and  ad  -  cd. 


QUESTIONS  ON  CHAPTER  II 

1.  Define  motion.     What  data  define  the  position  and  motion  of  a  point, 
line,  plane  figure  and  body,  the  latter  three  having  plane  motion? 
3 


34  THE  THEORY  OF  MACHINES 

2.  Two  observers  are  looking  at  the  same  object;  one  sees  it  move  while 
to  the  other  it  may  appear  stationary ;  explain  how  this  is  possible. 

3.  When  a  person  in  an  automobile,  which  is  gaining  on  a  street  car, 
looks  at  the  latter  without  looking  at  the  ground,  the  car  appears  to  be 
coming  toward  him.     Why? 

4.  Explain  the  difference  between  relative  and  absolute  motion  and  state 
the  propositions  referring  to  these  motions. 

6.  The  speeds  of  two  pulleys  are  100  revolutions  per  minute  in  the  clock- 
wise sense  and  125  revolutions  per  minute  in  the  opposite  sense,  respectively; 
what  is  the  relative  speed  of  the  former  to  the  latter? 

6.  In  a  geared  pump  the  pinion  makes  90  revolutions  per  minute  and  the 
pump  crankshaft  30  revolutions  in  opposite  sense;  what  is  the  motion  of  the 
pinion  relative  to  the  shaft? 

7.  Distinguish  between  the  instantaneous  and  complete  motion  of  a  body. 
What  information  gives  the  former  completely?     What  is  the  virtual  center? 

8.  What  is  the  virtual  center  of  a  wagon  wheel  (a)  with  regard  to  the 
earth,  (6)  with  regard  to  the  wagon? 

9.  A  vehicle  with  36-in.  wheels  is  moving  at  10  miles  an  hour;  what  are 
the  velocities  in  space  and  the  directions  of  motion  of  a  point  at  the  top  of 
the  tire  and  also  of  points  at  the  ends  of  a  horizontal  diameter?     Is  the 
motion  the  same  for  the  latter  two  points?     If  not,  find  their  relative  motion. 

10.  A  wheel  turns  at  150  revolutions  per  minute;  what  is  its  angular 
velocity  in  radians  per  second?     Also,  if  it  is  20  in.  diameter,  what  is  the 
linear  velocity  of  the  rim? 

11.  Give  the  information  necessary  to  locate  the  virtual  center  between 
two  bodies. 

12.  What  is  the  difference  between  the  virtual,  permanent  and  fixed 
center?     State  and  prove  the  theorem  of  the  three  centers. 

13.  Find  the  virtual  centers  for  the  stone  crusher  or  any  other  somewhat 
complicated  machine. 


/ 


\ 


CHAPTER  III 
VELOCITY  DIAGRAMS 

42.  Applications  of  Virtual  Center. — Some  of  the  main  appli- 
cations of  the  virtual  center  discussed  in  the  last  chapter  are  to 
the  determination  of  the  velocities  of  the  various  points  and  links 
in  mechanisms,  and  also  of  the  forces  acting  throughout  the 
mechanism  due  to  external  forces.     The  latter  question  will  be 
discussed  in  a  subsequent  chapter 'and  the  present  chapter  will 
be  confined  to  the  determination  of  velocities  and  to  the  repre- 
sentation of  these  velocities. 

43.  Linear  and  Angular  Velocities. — There  are  two  kinds  of 
velocities  which  are  required  in  machines,  the  linear  velocities  of 
the  various  points  and  the  angular  velocities  of  the  various  links, 
and  it  will  be  best  to  begin  with  the  determination  of  linear 
velocities. 

44.  Linear  Velocities  of  Points  in  Mechanisms. — The  linear 
velocities  of  the  various  points  may  be  required  in  one  of  two 
forms,  either  the  absolute  velocities  may  be  required  or  else  it 
may  be  only  desired  to  compare  the  velocities  of  two  points,  that 
is,  to  determine  their  relative  velocities.     The  latter  problem  may 
be  always  solved  without  knowing  the  velocity  of  any  point  in 
the  machine,  the  only  thing  necessary  being  the  shape  of  the 
mechanism  and  which  link  is  fixed,  while  for  the  determination 
of  the  absolute  velocity  of  a  point  in  a  mechanism  that  of  some 
point  or  link  must  be  known. 

Again,  the  two  points  to  be  compared  may  be  in  one  link,  or  in 
different  links,  and  the  solution  will  be  made  for  each  case  and  an 
effort  will  be  made  to  obtain  solutions  which  are  quite  general. 

45.  Points  in  the  Same  Link. — The  first  case  will  be  that  of  the 
four-link  mechanism,   frequently  referred    to,    containing   four 
turning  pairs  and  shown  in  Fig.  21,  and  the  letter  d  will  be  used 
to  indicate  the  fixed  link.     As  a  first  problem,  let  the  velocity 
of  any  point  A  i  in  a  be  given  and  that  of  another  point  A  2  in  the 
same  link  be  required.     The  six  virtual  centers  have  been  found 
and  marked  on  the  drawing,  and  the  link  a  has  been  selected  for 
the  first  example  because  it  has  a  permanent  center  which  is  ad. 

35 


36  THE  THEORY  OF  MACHINES. 

Now  the  velocity  of  AI  which  is  assumed  given,  is  the  absolute 
velocity,  that  is  the  velocity  with  regard  to  the  earth.  From  A  i 
lay  off  A  iE  in  any  direction  to  represent  the  known  velocity  of  A  i 
and  join  ad-E  and  produce  this  line  outward  to  meet  the  line 
A  zF,  parallel  to  AiE,  in  the  point  F.  Then  A2F  will  represent 
the  linear  velocity  of  A 2  on  the  same  scale  that  AiE  represents 
the  velocity  of  A\.  (It  is  assumed  in  this  construction  that  ad, 
A  i  and  A2  are  in  the  same  straight  line.)  The  reasoning  is 
simple,  for  a  turns  with  regard  to  the  earth  about  the  center  ad 
and  hence,  since  AI  and  A2  are  on  the  same  link,  their  linear 
velocities  are  directly  proportional  to  their  distances  from  ad 
(Sec.  35)  so  that, 

Linear  velocity  of  AI       ad  —  AI       A\E 
Linear  velocity  of  A 2       ad  —  Az 


be 


If  the  linear  velocity  of  A  i  is  given,  so  that  A\E  can  be  drawn 
to  scale,,  then  the  construction  gives  the  numerical  value  of  the 
velocity  of  A2,  but  if  the  velocity  of  A  i  is  not  given  then  the  above 
method  simply  gives  the  relative  velocities  of  AI  and  A 2. 

Next,  let  it  be  required  to  find  the  velocity  of  a  point  B2  in  b, 
Fig.  22,  the  velocity  of  ^i  in  the  same  link  being  given  and  let  d 
be  the  fixed  link  as  before.  Now  since  it  is  the  absolute  velocity 
of  BI  that  is  given,  the  first  point  is  to  find  the  center  of  bd  about 
which  b  is  turning  with  regard  to  the  earth.  The  velocity  of  B\ 
then  bears  the  same  relation  to  that  of  B%  as  the  respective  dis- 
tances of  these  points  from  bd,  or 

Linear  velocity  of  B\  _  bd  -  B\ 
Linear  velocity  of  Bz       bd  —  Bz 

It  is  then  only  necessary  to  get  a  simple  graphical  method  of 
obtaining  this  ratio  and  the  figure  indicates  one  way.     First, 


VELOCITY  DIAGRAMS  37 

with  center  bd  draw  arcs  of  circles  through  B\  and  Bz  cutting 
bd-  ab  in  B'i  and  B'2.  Then  if  B\G  be  drawn  in  any  direction 
to  represent  the  given  velocity  of  BI,  it  may  be  readily  shown  that 
B'zH  parallel  to  B'iG,  will  represent  the  linear  velocity  of  B2,  or 
the  ratio  of  B\G  to  B'^H  is  the  ratio  of  the  velocities  of  BI 
and  B2. 

The  only  difference  between  this  and  the  last  case  is  that  in  the 
former  case  the  center  ad  used  was  permanent,  whereas  in  this 
case  the  center  bd  used  is  a  virtual  center. 

46.  Points  in  Different  Links. — If  it  were  required  to  compare 
the  linear  velocity  of  the  point  AI  in  a  with  that  of  BI  in  6  the 
method  would  be  as  indicated  in  Fig.  22.  The  two  links  con- 
cerned are  a  and  b  and  d  is  the 
fixed  link  and  these  links  have 
the  three  centers  ad,  ab,  bd, 
all  on  one  line,  also  ab  is  a 
point  common  to  a  and  b, 
being  a  point  on  each  link. 
Treating  it  as  a  point  in  a, 
proceed  as  in  the  first  ex- 
ample to  find  its  velocity. 
Thus  set  off  A  iE  in  any  direc-  pIG>  22. 

tion  to  represent   the  linear 

velocity  of  AI,  then  ab-F  parallel  to  A\E  will  represent  the  ve- 
locity of  ab  to  the  same  scale.  Now  treat  ab  as  a  point  in  b  and 
its  velocity  is  given  as  ab-F,  so  that  the  matter  now  resolves 
itself  into  finding  the  velocity  of  a  point  BI  in  6,  the  velocity 
of  the  point  ab  in  the  same  link  being  given. 

It  must  again  be  remembered  that  ab  —  F  represents  the 
absolute  velocity  of  the  point  ab,  that  is,  the  velocity  of  this 
point,  using  the  fixed  frame  of  the  machine  as  the  standard.  With 
regard  to  the  frame  the  link  b  is  turning  about  the  center  bd,  thus 
for  the  instant  b  turns  relative  to  d  about  bd,  and  the  velocities 
of  all  points  in  it  at  this  instant  are  simply  proportional  to  their 
distances  from  bd.  The  velocity  of  B i  is  to  the  velocity  of  ab  in 
the  ratio  bd-'Bi  to  bd-ab,  and  in  order  to  get  this  ratio  con- 
veniently, draw  the  arc  BiB'i  with  center  bd,  then  join  bd-F 
and  draw  B\G  parallel  to  ab-F  to  meet  bd-F  in  G,  then  B'iG 
represents  the  velocity  of  B  i  in  the  link  b  on  the  same  scale  that 
AiE  represents  the  velocity  of  AI. 

Notice  that  in  dealing  with  the  various  links  in  finding  relative 


38 


THE  THEORY  OF  MACHINES 


velocities  it  is  .necessary  to  use  the  centers  of  the  links  under  con- 
sideration with  regard  to  the  fixed  link;  thus  the  centers  ad  and 
bd  and  the  common  center  ab  are  used.  The  reason  ad  and  bd 
are  employed,  is  because  the  velocities  under  consideration  are 
all  absolute. 

To  compare  the  velocity  of  any  point  A!  in  a  with  that  of  C\ 
in  c,  Fig.  23,  it  would  be  necessary  to  use  the  centers  ad,  ac  and 
cd.  Proceeding  as  in  the  former  case  the  velocity  of  ac  is  found 
by  drawing  the  arc  AiL  with  center  ad  and  drawing  LN  in  any 
direction  to  represent  the  velocity  of  A  i  on  a  chosen  scale,  than 
the  line  ac- M  parallel  to  LN  meeting  ad  —  N  produced  in  M 
will  represent  the  velocity  of  ac.  Join  cd  -  M,  and  draw  the  arc 
CiC'i  with  center  cd,  then  C'iK  parallel  to  ac-  M,  will  represent 
the  linear  velocity  of 


FIG.  23. 

A  general  proposition  may  be  stated  as  follows:  The  velocity 
of  any  point  A  in  link  a  being  given  to  find  the  velocity  of  F  in 
/,  the  fixed  link  being  d.  Find  the  centers  ad,  fd  and  af,  and 
using  ad  and  the  velocity  of  A,  find  the  velocity  of  af,  and  then 
treating  af  as  a  point  in  /  and  using  the  center  fd,  find  the  velocity 
ofF. 

47.  Relative  Angular  Velocities. — Similar  methods  to  the 
preceding  may  be  employed  for  finding  angular  velocities  in 
mechanisms. 

Let  any  body  having  plane  motion  turn  through  an  angle  8 

about  any  axis,  either  on  or  off  the  body,  in  time  t,  then  the 

/j 

angular  velocity  of  the  body  is  defined  by  the'  relation  «  =  .  •    As 

all  links  in  a  mechanism  move  except  the  fixed  link,  there  are  in 
general  as  many  different  angular  velocities  as  there  are  moving 
links.  The  angular  velocities  of  the  various  links  a,  b}  c}  etc., 


VELOCITY  DIAGRAMS  39 

will  be  designated  by  coa,  cot',  coc,  etc.,  respectively,  the  unit  being 
the  radian  per  second. 

As  in  the  case  of  linear  velocities,  angular  velocities  may  be 
expressed  either  as  a  ratio,  in  which  case  the  result  is  a  pure 
number,  or  as  a  number  of  radians  per  second,  the  method  de- 
pending on  the  kind  of  information  sought  and  also  upon  the 
data  given.  Unless  the  data  includes  the  absolute  angular 
velocity  of  one  link  it  is  quite  impossible  to  obtain  the  absolute 
velocity  of  any  other  link  and  it  is  only  the  ratio  between  these 
velocities  which  may  be  found. 

48.  Methods  of  Expressing  Velocities. — In  rinding  the  rela- 
tive angular  velocities  between  two  bodies  it  is  most  usual  to 

express  the  result  as  a  ratio,  thus  —  ,  which  result  is,  of  course,  a 

t  pure  number,  such  a  method  is  very  commonly  employed  in 
connection  with  gears,  pulleys  and  other  devices.  If  a  belt  con- 
nects two  pulleys  of  30  in.  and  20  in.  diameter  their  velocity  ratio 
will  be  2%o  =  %)  that  is,  when  standing  on  the  ground  and  count- 
ing the  revolutions  with  a  speed  counter,  one  of  the  wheels  will 
be  found  to  make  two-thirds  the  number  of  revolutions  the  other 
one  does,  and  this  ratio  is  alWiys  the  same  irrespective  of  the 
absolute  speed  of  either  pulley. 

It  happens,  however,  that  it  may  be  necessary  to  know  the 
relative  angular  velocities  in  a  different  form,  that  is,  it  may  be 
desired  to  know  how  fast  one  of  the  wheels  goes  considering  the 
other  as  a  standard ;  the  result  would  then  be  expressed  in  radians 
per  second.  Suppose  a  gear  a  turns  at  20  revolutions  per  minute, 
wa  =  2.09  radians  per  second,  and  meshes  with  a  gear  b  running 
at  30  revolutions  per  minute,  co6  =  3.14  radians  per  second,  the 
two  wheels  turning  in  opposite  sense,  then  the  velocity  of  a 
with  regard  to  b  is  co0  —  ub  =  2.09  —  (  —  3.14)  =  +5.23  radians 
per  second,  that  is,  if  one  stood  on  gear  6  and  looked  at  gear  a, 
the  latter  would  appear  to  turn  in  the  opposite  sense  to  b  and  at  a 
rate  of  5.23  radians  per  second.  If,  on  the  other  hand,  one  stood 
on  a,  then  dd,  since  co&  —  coa  =  —3.14  —2.09  =  —  5.23,  b  would 
appear  to  turn  backward  at  a  rate  of  5.23  radians  per  second,  the 
relative  motion  of  a  to  b  being  equal  and  opposite  to  that  of  b 
with  regard  to  a. 

The  first  method  of  reckoning  these  velocities  will  alone  be 
employed  in  this  discussion  and  the  construction  will  now  be 
explained. 


40 


THE  THEORY  OF  MACHINES 


49.  Relative  Velocities  of  Links. — Given  the  angular  velocity 
of  a  link  a  to  find  that  of  any  other  link  b.  Find  the  three  centers 
ad,  bd  and  ab,  then  as  a  point  in  a,  ab  has  the  linear  velocity 
(ad  —  ab)  coa  and  as  a  point  in  b,  ab  has  the  velocity  (bd  —  ab)  co&. 
But  as  ab  must  have  the  same  velocity  whether  considered  as  a 

point  in  a  or  in  b,  then  (ad  —  ab)  coa  =  (bd  —  ab)  ub,  or  --  _ 

C0a 

— .     The  construction  is  shown  in  Fig.  24  and  will  require 
bd  —  ab 

very  little  explanation.     Draw  a  circle  with  center  ab  and  radius 
ab  —  ad,  which  cuts  ab  —  bd  in  G,  lay  off  bd  —  F  in  any  direc- 


FIG.  24. 

tion  to  represent  co0  on  chosen  scale,  then  draw  GE  parallel  to 
bd  —  F  to  meet  ab  —  F  in  E,  and  GE  will  represent  the  angular 
velocity  of  b  or  ub  on  the  same  scale. 

Similar  processes  may  be  employed  for  the  other  links  b  and  c, 
and  no  further  discussion  of  the  point  will  be  given  here.  The 
general  constructions  are  very  similar  to  those  for  finding  linear 
velocities. 

As  in  the  case  of  the  linear  velocities  the  following  general 
method  may  be  conveniently  stated :  The  angular  velocity  of  any 
link  a  being  given  to  find  the  angular  velocity  of  a  link  /,  d  being 
the  fixed  link.  Find  the  centers  ad,  fd  and  af,  then  the  angular 
velocity  coa  is  to  «/  in  the  same  ratio  that/d  —  af  is  to  ad  —  af. 

50.  Discussion  on  the  Method. — Although  the  determination 
of  the  linear  and  angular  velocities  by  means  of  the  virtual  center 


VELOCITY  DIAGRAMS  41 

is  simple  enough  in  the  cases  just  considered,  yet  when  it  is  em- 
ployed in  practice  there  is  frequently  much  difficulty  in  getting 
convenient  constructions.  Many  of  the  lines  locating  virtual 
centers  are  nearly  parallel  and  do  not  intersect  within  the  limits 
of  the  drafting  board,  and  hence  special  and  often  troublesome 
methods  must  be  employed  to  bring  the  constructions  within 
ordinary  bounds.  Further,  although  it  is  common  to  have 
given  the  motion  of  one  link  such  as  a,  and  often  only  the 
motion  of  one  other  point  or  link  say  /,  elsewhere  in  the  mech- 
anism is  desired,  requiring  the  finding  of  only  three  virtual 
centers,  ad,  af  and  df,  yet  frequently  in  practice  these  cannot  be 
obtained  without  locating  almost  all  the  other  virtual  centers 
in  the  mechanism  first.  This  may  involve  an  immense  amount  of 
labor  and  patience,  and  in  some  cases  makes  the  method 
unworkable. 

51.  Application  to  a  Mechanism. — A  practical  example  of  a 
more  complicated  mechanism  in  common  use  will  be  worked  out 
here  to  illustrate  the  method,  only  two  more  centers  af  and  bf 
being  found  than  those  necessary  for  the  solution  of  the  problem. 
Fig.  25  shows  the  Joy  valve  gear  as  frequently  used  on  locomotives 
and  other  reversing  engines,  more  especially  in  England:  a  rep- 
resents the  engine  crank,  b  the  connecting  rod,  and  c  the  piston, 
etc.,  as  in  the  ordinary  engine,  the  frame  being  d.  One  end  of  a 
link  e  is  connected  to  the  rod  b  and  the  other  end  to  a  link  /, 
the  latter  link  being  also  connected  to  the  engine  frame,  while  to 
the  link  e  a  rod  g  is  jointed,  which  rod  is  also  jointed  to  a  sliding 
block  h,  and  at  its  extreme  upper  end  to  the  slide  valve  stem  V. 
The  part  ra  on  which  h  slides  is  controlled  in  direction  by  the 
engineer  who  moves  it  into  the  position  shown  or  else  into  the 
the  dotted  position,  according  to  the  sense  of  rotation  desired  in 
the  crankshaft,  but  once  this  piece  m  is  set,  it  is  left  stationary 
and  virtually  becomes  fixed  for  the  time. 

A  very  useful  problem  in  such  a  case  is  to  find  the  velocity  of 
the  valve  and  stem  V  for  a  given  position  and  speed  of  the  crank- 
shaft. The  problem  concerns  three  links,  a,  d  and  g,  the  upper 
end  of  the  latter  link  giving  the  valve  stem  its  motion,  so  that 
the  three  centers  ad,  ag,  and  dg  are  required.  First  write  on  all 
the  centers  which  it  is  possible  to  find  by  inspection,  such  as  ad, 
ab,  be,  be,  cd,  ac,  ef,  etc.,  and  then  proceed  to  find  the  required 
centers  by  the  theorem  of  three  centers  given  in  Sec.  39.  The 
centers  ag  and  dg  cannot  be  found  at  once  and  it  will  simplify 


42 


THE  THEORY  OF  MACHINES 


the  work  to  set  down  roughly  in  a  circle  (not  necessarily  accur- 
ately) anywhere  on  the  sheet  points  which  are  approximately 
equidistant,  there  being  one  point  for  each  link,  in  this  case  eight. 
Now  letter  these  points  a,  b,  c,  d,  e,  f,  g,  h,  to  correspond  with  the 
links.  As  a  center  such  as  ab  is  found  join  the  points  a  and  b  in 
the  lower  diagram  and  it  is  possible  to  join  a  fairly  large  number 


be 


\ 


ab 


ad 


af 


FIG.  25. — Joy  valve  gear. 

of  the  points  at  once,  then  any  two  points  not  joined  will  repre- 
sent a  center  still  to  be  found.  The  figure  shows  by  the  plain 
lines  the  stage  of  the  problem  after  the  centers  ab,  ac,  ad,  ae,  af, 
ag,  be,  bd,  df,  be,  cd,  de,  df,  dg,  dh,  ef,  eg,  gh,  have  been  found, 
which  represents  the  work  necessary  to  find  the  above  three 
centers  ag,  ad  and  dg. 

When  all  points  on  the  lower  figure  are  joined,  all  the  centers 


VELOCITY  DIAGRAMS  43 

have  been  found,  and  the  figure  shows  by  inspection  what  centers 
can  be  found  at  any  time,  for  it  is  possible  to  find  any  center 
provided  there  are  two  paths  between  the  two  points  correspond- 
ing to  the  center.  It  may  happen  that  there  will  appear  to  be 
two  paths  between  a  given  pair  of  points,  but  on  examination  it 
may  be  found  that  the  paths  are  really  coincident  lines,  in  which 
case  they  will  not  fix  the  center  and  another  path  is  necessary. 
The  lower  diagram  in  Fig.  25  shows  that  the  centers  ab,  ac,  ad,  de, 
df,  are  known,  while  the  centers  ah,  bg,  ch,  are  not  known, 
and  that  the  center  fg  can  probably  be  found  as  there  are  the  two 
paths  fa  —  ag  and  fd  —  dg  between  them  as  well  as  the  path  fe  — 
eg.  The  center  gc  could  not,  however,  be  found  before  gd}  as 
there  would  then  be  only  one  path  ga  —  ac  between  the  points. 

Having  now  found  the  centers  ad,  dg,  and  ag,  proceed  as  in  the 
previous  cases  to  find  the  velocity  of  V  or  the  valve  from  the 
known  velocity  of  a.  If  the  velocity  of  the  crankpin  ab  is  given, 
revolve  ad  —  ab  into  the  line  ad  —  dg  and  lay  off  a'b'  —  B 
to  represent  the  velocity  of  ab  on  any  scale.  Join  ad  —  B,  then 
ag  —  A  parallel  to  a'b'  —  B  gives  the  velocity  of  ag.  Next  join 
dg  —  A  and  with  center  dg  draw  the  arc  VV,  then  V'C  parallel 
to  ag  —  A  will  represent  the  velocity  of  the  valve  V.  The  whole 
process  is  evidently  very  cumbersome  and  laborious  and  is  fre- 
quently too  lengthy  to  be  adopted.  The  reader's  attention  is 
called  to  the  solution  of  the  same  problem  by  a  different  method 
in  Chapter  IV. 

52.  It  must  not  be  assumed  that  the  methods  here  described 
are  not  used,  because,  in  spite  of  the  labor  involved  they  are 
frequently  more  simple  than  any  other  method  and  a  number  of 
applications  of  the  virtual  center  are  given  farther  on  in  the 
present  treatise.     It  is  always  necessary  to  do  whatever  work  is 
required  in  solving  problems,  the  importance  of  which  frequently 
justifies  large  expenditure  of  time.     In  many  cases  the  method 
described  in  the  next  chapter  simplifies  the  work,  and  the  reader's 
judgment  will  tell  him  in  each  case  which  method  is  likely  to 
suit  best. 

53.  Graphical  Representation  of  Velocities. — It  is  frequently 
desirable  to  have  a  diagram  to  represent  the  velocities  of  the 
various  points  in  a  machine  for  one  of  its  complete  cycles,  as  the 
study  of  such  diagrams  gives  very  much  information  about  the 
nature  of  the  machine  and  of  the  forces  acting  on  it.     Two 


44 


THE  THEORY  OF  MACHINES 


methods  are  in  fairly  common  use  (1)  by  means  of  a  polar  dia- 
gram, (2)  by  means  of  a  diagram  on  a  straight  base. 

To  illustrate  these  a  very  simple  case,  the  slider-crank 
mechanism,  Fig.  26,  will  be  selected,  and  the  linear  velocities  of 
the  piston  will  be  determined,  a  problem  which  may  be  very 
conveniently  solved  by  the  method  of  virtual  centers.  Let 
the  speed  of  the  engine  be  known,  and  calculate  the  linear  velocity 
of  the  crankpin  ab;  for  example,  let  a  be  5  in.  long,  and  let  the 


FIG.  26. — Piston  velocities. 

speed  be  300  revolutions  per  minute,  then  the  velocity  of  the 

300        5 
crankpin  =  2-jr  X  ^TT  X  T^  =  13.1  ft.  per  second.     Now  be  is  a 

point  on  both  the  piston  c  and  on  the  rod  b  and  clearly  the  velocity 
of  be  is  the  same  as  that  of  c,  the  latter  link  having  only  a  motion 
of  translation,  and  further,  the  velocity  of  the  crankpin  ab  is 
known,  which  is  also  the  same  as  that  of  the  forward  end  of  the 
connecting  rod.  The  problem  then  is:  given  the  velocity  of  a 
point  ab  in  b  to  find  the  velocity  of  be  in  the  same  link,  and  from 
what  has  already  been  said  (Sec.  35),  the  relation  may  be  written: 

velocity  of  piston         velocity  of  be       bd  —  be 
velocity  of  crankpin  ~~  velocity  of  ab  ~  bd  —  ab 


VELOCITY  DIAGRAMS  45 

But  by  similar  triangles 

bd  —  be  _  ac  —  ad 
bd  —  ab  ~  ad  —  ab 
so  that 

velocity  of  piston     _  ad  —  ac 
velocity  of  crankpin       ad  —  ab 

and  as  ad  —  ab  is  constant  for  all  positions  of  the  machine,  it  is 
evident  that  ad  —  ac  represents  the  velocity  of  the  piston  on  the 
same  scale  as  the  length  of  a  represents  the  linear  velocity  of  the 
crankpin.  Or,  in  the  case  chosen,  if  the  mechanism  is  drawn  full 
size  then  ad  —  ab  =  5  in.,  and  the  scale  will  be  5  in.  •=  13.1  ft. 
per  second  or  1  in.  =  2.62  ft.  per  second. 

54.  Polar  Diagram. — Now  it  is  convenient  to  plot  this  velocity 
of  the  piston  either  along  a  as  ad  —  E  if  the  diagram  is  to  show 
the  result  for  the  different  crank  positions,  or  vertically  above  the 
piston  as  be  —  F,  if  it  is  desired  to  represent  the  velocity  for 
different  positions  of  the  piston.     If  this  determination  for  the 
complete  revolution  is  made,  there  are  obtained  the  two  diagrams 
shown,  the  one  OEGOHJO  is  called  a  polar  diagram,  0  being  the 
pole.     The  diagram  consists  of  two  closed  curves  passing  through 
0  and  both  curves  are  similar;  in  fact  the  lower  one  can  be'  ob- 
tained from  the  upper  by  making  a  tracing  of  the  latter  and  turn- 
ing it  over  the  horizontal  line  ad  —  be.     The  longer  the  connect- 
ing rod  the  more  nearly  are  the  curves  symmetrical  about  the 
vertical  through  0,  and  for  an  infinitely  long  rod  the  curves  are 
circles,  tangent  to  the  horizontal  line  at  0. 

55.  Diagram  on  a  Straight  Base. — The  diagram  found  by 
laying  off  the  velocities  above  and  below  the  piston  positions  is 
KFLMK,  and,  as  the  figure  shows,  is  egg-shaped  with  the  small 
end  of  the  egg  toward  0,  and  the  whole  curve  symmetrical  about 
the  line  of  travel  of  the  piston.     Increasing  the  length  of  the  rod 
makes  the  curve  more  nearly  elliptical,  and  with  the  infinitely 
long  rod  it  is  a  true  ellipse. 

If  the  direction  of  motion  of  the  piston  does  not  pass  through 
ad,  then  the  curve  FKML  is  not  symmetrical  about  the  line  of 
motion  of  the  piston,  but  takes  the  form  shown  at  Fig.  27,  where 
the  piston's  direction  passes  above  ad,  a  device  in  which  it  is 
clear  from  the  velocity  diagram  that  the  mean  velocity  of  the 
piston  on  its  return  stroke  is  greater  than  on  the  out  stroke,  and 
which  may,  therefore,  be  used  as  a  quick-return  motion  in  a  shaper 


46  THE  THEORY  OF  MACHINES 

or  other  similar  machine.  Engines  are  sometimes  made  in  this 
way,  but  with  the  cylinders  only  slightly  offset,  and  not  as  much 
as  shown  in  the  figure. 

56.  Pump  Discharge. — One  very  useful  application  of  such 
diagrams  as  those  just  described  may  be  found  in  the  case  of 
pumping  engines.  Let  A  be  the  area  of  the  pump  cylinder  in 
square  feet,  and  let  the  velocity  of  the  plunger  or  piston  in  a 
given  position  be  v  ft.  per  second,  as  found  by  the  preceding 
method,  let  Q  cu.  ft.  per  second  be  the  rate  at  which  the  pump 
is  discharging  water  at  any  instant,  then  evidently  Q  =  Av  and 
as  A  is  the  same  for  all  piston  positions,  Q  is  simply  proportional 


FIG.  27. — Off-set  cylinder. 

to  v,  or  the  height  of  the  piston  velocity  diagram  represents 
the  rate  of  delivery  of  the  pump  for  the  corresponding  piston 
position. 

If  a  pipe  were  connected  directly  to  the  cylinder,  the  water  in 
it  would  vary  in  velocity  in  the  way  shown  in  the  velocity  dia- 
gram (a),  Fig.  28,  the  heights  on  this  diagram  representing  piston 
velocities  and  hence  velocities  in  the  pipe,  while  horizontal  dis- 
tances show  the  distances  traversed  by  the  piston.  The  effect 
of  both  ends  of  a  double-acting  pump  is  shown;  this  variation  in 
velocity  would  produce  so  much  shock  on  the  pipe  as  to  injure 
it  and  hence  a  large  air  chamber  would  be  put  on  to  equalize  the 
velocity. 

Curve  (6)  shows  two  pumps  delivering  into  the  same  pipe, 
their  cranks  being  90°  apart,  the  heavy  line  showing  that  the 
variation  of  velocity  in  the  pipe  line  is  less  than  before  and  re- 
quires a  much  smaller  air  chamber.  At  (c)  is  shown  a  diagram 
corresponding  to  three  cranks  at  120°  or  a  three-throw  pump, 


VELOCITY  DIAGRAMS 


47 


in  which  case  the  variation  in  velocity  in  the  pipe  line  would  be 
much  smaller  still,  this  velocity  being  represented  by  the  height 
up  to  the  heavy  line.  All  the  curves  are  drawn  for  the  case  of  a 
very  long  connecting  rod,  or  of  a  pump  like  Fig.  6. 

Thus  the  velocity  diagram  enables  the  study  of  such  a  problem 
to  be  made  very  accurately,  and  there  are  many  other  useful 
purposes  to  which  it  may  be  put,  and  which  will  appear  in  the 
course  of  the  engineer's  experience.  Angular  velocities  may,,  of 
course,  be  plotted  the  same  way  as  linear  velocities. 


&th  Stcoke  ^     ^ 


(a)    Single  Cylinder  Pump 


Piston  Positions 


Resultant  Velocity 


(6)  Two  Cylinders.  Cranks  at  90 


Resultant  Velocity 


^YYYYYtYYYYYYYYA 


vVV 


(c)    Three  Cylinders,  Cranks  at  120° 
FIG.  28. — Rate  of  discharge  from  pumps. 

Another  method  of  finding  both  linear  and  angular  velocities 
is  described  in  the  next  chapter,  and  a  few  suggestions  are  made 
as  to  further  uses  of  these  velocities  in  practice. 


QUESTIONS  ON  CHAPTER  III 

1.  In  the  mechanism  of  Fig.  21,  a,  b,  c  and  d  are  respectively  3,  15,  10  and 
18  in.  long,  d  is  fixed  and  a  turns  at  160  revolutions  per  minute.     Find  the 
velocity  of  the  center  of  each  link  with  a  at  45°  to  d. 

2.  Find  also  the  angular  velocities  of  the  links  in  the  same  case  as  above. 

3.  JAn  8-in  by  10-in.  engine  has  a  connecting  rod  20  in.  long  and  a  speed  / 
of  250  revolutions  per  minute.     Find  the  velocity  of  the  center  of  the  rod  and*, 
the  angular  velocity  of  the  latter  in  radians  per  second  (a)  for  the  dead 
points,  (6)  when  the  crank  has  moved  45°  from  the  inner  dead  point,  (c)  for 
90°  and  135°  crank  angles. 


48  THE  THEORY  OF  MACHINES 

4.  Find  the  linear  velocity  of  the  left  end  of  the  jaw  in  Fig.  84,  knowing 
the  angular  velocity  of  the  camshaft. 

6.  Plot  a  diagram  on  a  straight  base  for  every  15°  of  crank  angle  for 
the  piston  and  center  of  the  rod  in  question  3;  also  plot  a  polar  diagram  of 
the  angular  velocity  of  the  rod. 

6.  In  the  mechanism  of  Fig.  27  with  a  =  6,  b  =  18  in.,  and  the  line  of 
travel  of  c,  9  in.  above  ad,  plot  the  velocity  diagram  for  c. 

7.  What  maximum   speed  will  be  obtained  with  a   Whitworth   quick- 
return  motion,  Fig.  37,  with  d  =  %  in.,  a  =  2  in.,  6  =  2%  in.  and  e  =  13 
in.,  the  line  of  motion  of  the  table  passing  through  ad  ? 

8.  Find  the  velocity  of  the  tool  in  one  of  the  riveters  given  in  Chapter  IX, 
assuming  the  velocity  of  the  piston  at  the  instant  to  be  known. 

9.  Draw  the  polar  diagram  for  the  angular  velocity  of  the  valve  steam  in 
the  mechanism  of  Fig.  40. 


CHAPTER  IV 
THE  MOTION  DIAGRAM 

57.  Uses   of   Velocity   Diagrams. — In   the   previous   chapter 
something  has  been  said  about  the  methods  of  finding  the  veloci- 
ties of  various   points   and   links   in   mechanisms,    and  a  few 
applications  of  the  methods  were  given.     As  a  matter  of  fact 
there  are  a  very  great  number  of  cases  in  which  such  velocity 
diagrams  are  of  great  value  in  studying  the  conditions  existing 
in  machines.     Such  problems,  for  example,  as  the  value  of  a  quick- 
return  motion,  or  of  a  given  type  of  valve  gear  or  link  motion; 
or  again,  problems  involving  the  action  of  forces  in  machines, 
such  as  the  turning  moment  produced  on  the  crankshaft  by  a 
given  piston  pressure,  or  the  belt  pull  necessary  to  crush  a  cer- 
tain kind  of  stone  in  a  stone  crusher,  and  many  other  similar 
matters. 

All  of  the  above  problems  may  be  solved  by  the  determination 
of  the  velocities  of  various  parts  and  hence  the  matter  of  finding 
these  velocities  deserves  some  further  consideration,  more  espe- 
cially in  view  of  the  fact  that  a  somewhat  simpler  method  than 
that  described  in  Chapter  III,  and  which  enables  the  rapid  solu- 
tion of  all  such  problems,  is  available  and  may  now  be  discussed. 
The  graphical  method  of  solution  is  usually  the  best,  because  it 
is  simple  and  because  the  designing  engineer  always  has  drafting 
instruments  available  for  such  a  method,  and  further  because 
motions  in  machines  are  frequently  so  complex  as  to  render 
mathematical  solutions  altogether  too  cumbersome. 

58.  Method  to  be  Used. — In  all  machines  there  is  one  part 
which  has  a  definitely  known  motion,  and  frequently  this  motion 
is  one  of  rotation  about  a  fixed  center  at  uniform  speed,  as  in  the 
case  of  the  flywheel  of  an  engine,  or  the  belt  wheel  of  a  stone 
crusher  on  punch  or  machine  tool,  this  part  is  called  the  link  of    / 
reference.     Provided  the  motion   of  this  link  is  known,  it  is 
possible  to  definitely  determine  the  motions  of  all  other  parts, 
but  if  its  motion  is  not  known,  then  all  that  is  possible  is  the 
determination  of  the  relative  motions  of  the  various  parts;  the 
method  described  here  may  be  used  in  either  case. 

4  49 


50 


THE  THEORY  OF  MACHINES 


The  construction  about  to  be  explained  has  been  called  by  its 
discoverer1  the  phorograph  method,  and,  as  the  name  suggests, 
is  a  method  for  graphically  representing  the  motions.  It  is  a 
vector  method  of  a  kind  similar  to  that  used  in  determining  the 
stresses  in  bridges  and  roofs  with  the  important  differences  that 
the  vector  used  in  representing  stresses  are  always  parallel  to  the 
member  affected,  while  the  vector  representing  velocity  is  in 
many  cases  normal  to  the  direction  of  the  link  concerned  and 

further  that  the  diagram  is 
drawn  on  an  arbitrarily  selected 
Jink  of  reference  which  is  itself 
moving. 

59.  The  Phorograph. — The 
phorograph  is  a  construction 
by  which  the  motions  of  all 
points  on  a  machine  may  be 
represented  in  a  convenient 
graphical  manner.  As  dis- 
cussed here  the  only  applica- 
tion made  is  to  plane  motion 
although  the  construction  may 
readily  be  modified  so  as  to 

make  it  apply  to  non-plane  motion,  but  in  most  cases  of  the 
latter  kind  any  graphical  construction  becomes  complicated. 
The  method  is  based  pn  very  few  important  principles  and 
these  will  first  be  explained. 

60.  Relative  Motion  of  Points  in  Bodies.  First  Principle. — The 
first  principle  is  that  any  one  point  in  a  rigid  body  can  move 
relatively  to  any  other  point  in  the  same  body  only  in  a  direc- 
tion at  right  angles  to  the  straight  line  joining  them;  that  is  to 
say,  if  the  whole  body  moves  from  one  position  to  another,  then 
the  only  motion  which  the  one  point  has  that  the  other  has  not  is 
in  a  direction  normal  to  the  line  joining  them.  To  illustrate 
this  take  the  connecting  rod  of  an  engine,  a  part  of  which  is  shown 
at  a  in  Fig.  29  and  let  the  two  points  B  and  C,  and  hence  the 
line  BC,  lie  in  the  plane  of  the  paper.  Let  the  rod  move  from 
a  to  a',  the  line  taking  up  the  corresponding  new  position  B'C'. 
During  the  motion  above  described  C  has  moved  to  C'  and  B 

1  PROFESSOR  T.  R.  ROSEBRUQH  of  the  University  of  Toronto  discovered 
the  method  and  first  gave  it  to  his  classes  about  25  years  ago,  but  the 
principle  has  not  appeared  in  print  before. 


FIG.  29. 


THE  MOTION  DIAGRAM  51 

to  B'.  Now  draw  BBl}  parallel  to  CC'  and  C'Bl}  parallel  to  BC, 
then  an  inspection  of  the  figure  shows  at  once  that  if  the  rod  had 
only  moved  to  B\C'  the  points  B  and  C  would  have  had  exactly 
the  same  net  motion,  that  is,  one  of  translation  through  CC'  = 
BBi  in  the  same  direction  and  sense,  and  hence  B  and  C  would 
have  had  no  relative  motion.  But  when  the  rod  has  moved  to 
a,  B  has  had  a  further  motion  which  C  has  not  had,  namely 

#!#'. 

Thus  during  the  motion  of  a,  B  has  had  only  one  motion  not 
shared  by  C,  or  B  has  moved  relatively  to  C  through  the  arc 
BiB',  and  at  each  stage  of  the  motion  the  direction  of  this  arc 
was  evidently  at  right  angles  to  the  radius  from  C',  or  at  right 
angles  to  the  line  joining  B  and  C. 

Thus  when  a  body  has  plane  motion  any  point  in  the  body 
can  move  relatively  to  any  other  point  in  the  body  only  at  right 
angles  to  the  line  joining  the  two  points.  It  follows  from  this 
that  if  the  line  joining  the  two  points  should  be  normal  to  the 
plane  of  motion,  then  the  two  points  could  have  no  relative 
motion. 

61.  Second  Principle. — Let  Fig.  30  represent  a  mechanism 
having  four  links,  a,  b,  c  and  d,  joined  together  by  four  turning 


FIG.  30. 

pairs  0,  P,  Q  and  R  as  indicated.  This  mechanism  is  selected 
because  of  its  common  application  and  the  reader  will  find  it 
used  in  many  complicated  mechanisms.  For  example,  it  forms 
half  the  mechanism  used  in  the  beam  engine,  when  the  links  are 
somewhat  differently  proportioned,  a  being  the  crank,  6  the  con- 
necting rod  and  c  one-half  of  the  walking  beam.  The  same  chain 
is  also  used  in  the  stone  crusher  shown  at  Fig.  95  and  in  many 
other  places. 

The  second  principle  upon  which  the  phorograph  depends  may 
now  be  explained  by  illustrating  with  the  above  mechanism. 


52  THE  THEORY  OF  MACHINES 

In  this  mechanism  the  fixed  link  is  d  which  will  be  briefly 
referred  to  as  the  frame.  Thus  0  and  R  are  fixed  bearings  or 
permanent  centers,  while  P  and  Q  move  in  arcs  of  circles  about 
O  and  R  respectively.  Choose  one  of  the  links  as  the  link  of 
reference,  usually  a  or  c  will  be  most  suitable  as  they  both  have 
a  permanent  center  while  b  has  not;  the  one  actually  selected  is 
a.  Imagine  that  to  a  there  is  attached  an  immense  sheet  of 
cardboard  extending  indefinitely  in  all  directions  from  O,  and 
for  brevity  the  whole  sheet  will  be  referred  to  as  a. 

A  consideration  of  the  matter  will  show  that  on  the  cardboard 
on  the  link  a  there  are  points  having  all  conceivable  motions  and 
velocities  in  magnitude,  direction  and  sense.  Thus,  if  a  circle 
be  drawn  on  a  with  center  at  0,  all  points  on  the  circle  will  have 
velocities  of  the  same  magnitude,  but  the  direction  and  sense  will 
be  different;  or  if  a  vertical  line  be  drawn  through  O}  all  points 
on  this  line  will  move  in  the  same  direction,  that  is,  horizontal, 
those  above  0  moving  in  opposite  sense  to  those  below  and  all 
points  having  different  velocities.  If  any  point  on  a  be  selected, 
its  velocity  will  depend  on  its  distance  from  0,  the  direction  of  its 
motion  will  be  normal  to  the  radial  line  joining  it  to  0,  and  its 
sense  will  depend  upon  the  relative  positions  of  the  point  and 
0  on  the  radial  line..  The  above  statements  are  true  whether  a 
has  constant  angular  velocity  or  not,  and  are  also  true  even 
though  0  moves. 

From  the  foregoing  it  follows  that  it  will  be  possible  to 
find  a  point  on  a  having  the  same  motion  as  that  of  any  point  Q 
in  any  link  or  part  of  the  machine,  which  motion  it  is  desired  to 
study;  and  thus  to  collect  on  a  a  set  of  points  each  representing 
the  motion  of  a  given  point  on  the  machine  at  the  given  instant. 
Since  the  points  above  described  are  all  on  the  link  a,  their 
relative  motions  will  be  easily  determined,  and  this  therefore 
affords  a  very  direct  method  of  comparing  the  velocities  of  the 
various  points  and  links  at  a  given  instant.  If  the  motion  of  a 
is  known,  as  is  frequently  the  case,  then  the  motions  of  all  points 
on  a  are  known,  and  hence  the  motion  of  any  point  in  the  mechan- 
ism to  which  the  determined  point  on  a  corresponds;  whereas, 
if  the  motion  of  a  is  unknown,  only  the  relative  motions  of  the 
different  points  at  the  instant  are  known. 

A  collection  of  points  on  a  certain  link,  arbitrarily  chosen  as 
the  link  of  reference,  which  points  have  the  same  motions  as 
points  on  the  mechanism  to  which  they  correspond,  and  about 


THE  MOTION  DIAGRAM  53 

which  information  is  desired,  is  called  the  photograph  of  the 
mechanism,  because  it  represents  graphically  (vectorially)  the 
relative  motions  of  the  different  points  in  the  mechanism. 

62.  Third  Principle. — The  third  point  upon  which  this  graph- 
ical method  depends  is  that  the  very  construction  of  the  mechan- 
ism supplies  the  information  necessary  for  finding  in  a  simple 
way  the  representative  point  on  the  reference  link  corresponding 
to  a  given  point  on  the  mechanism;  this  representative  point  may 
be  conveniently  called  the  image  of  the  actual  point.  Looking  at 
the  mechanism  of  Fig.  30,  and  remembering  the  first  principle 
as  enunciated  in  Sec.  60,  it  is  clear  that  if  it  is  desired  to  study  the 
motion  and  velocity  of  such  a  point  as  Q,  then  the  mechanism 
gives  the  following  information  at  once: 

1.  The  motion  of  Q  relative  to  P  is  normal  to  QP  since  P 
and  Q  are  both  in  the  link  b,  and  as  P  is  also  a  point  in  link  a, 


the  motion  of  Q  ~  P  in  a  is  normal  to  QP,  or  the  motion  of  Q 
in  b  with  regard  to  a  point  P  in  a  is  known. 

2.  Since  Q  is  also  a  point  in  c  the  motion  of  Q  ~  R  is  normal  to 
QR.  But  R  is  a  point  in  the  fixed  link  d  and  hence  R  is  stationary 
as  0  is,  so  that  the  motion  of  Q  ~  R  is  the  same  as  the  motion 
of  Q  ~  0.  Hence  the  motion  of  Q  with  regard  to  a  second  point 
in  a  is  known. 

As  will  now  be  shown  these  facts  are  sufficient  to  determine  on 
a  point  Q',  a  having  the  same  motion  as  Q  and  the  method  of 
doing  this  will  now  be  demonstrated  in  a  general  way. 

63.  Images  of  Points. — Let  there  be  a  body  K,  Fig.  31,  con- 
taining two  points  E  and  F,  and  let  K  have  plane  motion  of  any 


54  THE  THEORY  OF  MACHINES 

nature  whatsoever,  the  exact  nature  of  its  motion  being  at  pres- 
ent unknown.  On  some  other  body  there  is  a  point  G  also 
moving  in  the  same  plane  as  K\  the  location  of  G  is  unknown 
and  the  only  information  given  about  it  is  that  its  instantaneous 
motion  relative  to  E  is  in  the  direction  G  —  1  and  its  motion 
relative  to  F  is  in  the  direction  G  —  2.  It  is  required  to  find  a 
point  G'  on  K  which  has  the  same  motion  as  (?;  the  point  Gf  is 
called  the  image  of  G. 

Referring  to  the  first  principle  it  is  seen  that  the  motion  of  any 
point  in  K  ~  E  is  perpendicular  to  the  lir*e  joining  this  point  to 
E,  for  example  the  motion  of  F  ~  E  is  perpendicular  to  FE.  Now 
a  point  G'  is  to  be  found  in  K  having  the  same  motion  as  G,  and  as 
the  direction  of  motion  of  G  ~  E  is  given,  this  gives  at  once  the 
position  of  the  line  joining  E  to  the  required  point;  it  must  be  per- 
pendicular to  G  —  1  and  pass  through  E.  The  point  could  not 
lie  at  H  for  instance,  because  then  'the  direction  of  motion  would 
be  perpendicular  to  HE,  which  is  different  to  the  specified  direc- 
tion G  —  1.  Thus  Gf  lies  on  a  line  EG'  perpendicular  to  G  —  1 
through  E. 

Similarly  it  may  be  shown  that  G'  must  lie  on  a  line  through 
F  perpendicular  to  G  —  2,  and  hence  it  must  lie  at  the  intersec- 
tion of  the  lines  through  E  and  F  or  at  G'  as  shown  in  Fig.  31. 
Then  G'  is  a  point  on  K  having  the  same  motion  as  G  in  some 
external  body. 

64.  Possible  Data. — A  little  consideration  will  show  that  it  is 
not  possible  to  assume  at  random  the  sense  or  magnitude  of  the 
motions,  but  only  the  two  directions.  The  point  G'  could,  how- 
ever, be  found  by  assuming  the  data  in  another  form;  for  example, 
if  the  angular  velocity  of  K  were  known  and  also  the  magnitude 
direction  and  sense  of  motion  of  G  ~  E,  G'  could  be  located,  and 
then  the  motion  of  G~F  could  be  determined,  the  reader  will 
readily. see  how  this  is  done.  In  general  the  data  is  given  in 
the  form  stated  first,  as  will  appear  later. 

Now  as  to  the  information  given,  the  discussion  farther  on  will 
show  this  more  clearly  but  to  introduce  the  subject  in  a  simple 
way  let  the  virtual  center  of  the  body  at  the  given  instant,  with 
regard  to  the  earth,  be  H  and  let  the  body  be  rotating  at  this 
instant  in  a  clockwise  sense  at  the  rate  of  n  revolutions  per  min- 
ute or  co  =  -^^radians  per  second.  Then  the  motion  of  G  in 
space  is  in  the  direction  normal  to  G'H,  and  it  is  moving  to  the 


THE  MOTION  DIAGRAM  55 

right  with  a  velocity  G'H.u  ft.  per  second,  where  G'H  is  in  feet. 
Further,  the  motion  of  G  ~  E  is  in  the  sense  (7—1  and  the  veloc- 
ity of  G  ~  E  is  EG'.u  ft.  per  second. 

65.  Application  to  Mechanisms. — The  application  of  the  above 
.  principles  to  the  solution  of  problems  in  machinery  will  illustrate 
the  method  very  well,  and  in  doing  this'  the  principles  upon  which 
the  construction  depends  should  be  carefully  studied,  and  atten- 
tion paid  to  the  fact  that  if  too  much  is  assumed  the  different 
items  may  not  be  consistent. 

The  simple  mechanism  with  four  links  and  four  turning  pairs 
will  be  again  selected  as  the  first  example,  and  is  shown  in  Fig. 
32,  the  letters  a,  6,  c,  d,  0,  P,  Q  and  R  having  the  same  significance 
as  in  former  figures  and  a  is  chosen  as  the  link  of  reference,  or 
more  conveniently,  the  primary  link,  a  rough  outline  being  shown 
to  indicate  its  wide  extent.  In  future  this  outline  will  be  omitted. 
It  is  required  to  find  the  linear  velocities  of  the  point  S  the  center 
of  6,  of  T  in  c  and  of  Q,  also  the  angular  velocities  of  b  and  c  com- 
pared to  a  while  the  mechanism  is  passing  through  the  position 
shown  in  the  figure. 

Points  will  first  be  found  on  a  having  the  same  motions  as  Q 
and  R}  these  points  being  the  images  of  Q  and  R,  and  are  indi- 
cated by  accents;  thus  Qf  is  a  point  on  a  having  the  same  motion 
in  every  respect  as  Q  actually  has. 

Inspection  at  once  shows  that  since  P  is  a  point  in  a  therefore 
P1  the  image  of  P  will  coincide  with  the  latter,  and  if  w  be  the 
angular  velocity  of  a  (where  co  may  be  constant  or  variable), 
then  the  linear  velocity  of  P  at  the  instant  is  OP.u  =  aco  ft.  per 
second,  where  a  is  the  length  in  feet  of  the  link  a.  The  direc- 
tion of  motion  of  P  is  perpendicular  to  OP  and  its  sense  must 
correspond  with  co.  Such  being  the  case,  the  length  OP  or  a 
represents  aco  ft.  per  second,  so  that  the  velocity  scale  iso>:l. 
Again  since  R  is  stationary  it  is  essential  that  R'  be  located  at 
O  the  only  stationary  point  in  the  link  a. 

The  point  Q'  may  be  found  thus:  The  direction  of  motion 
of  Q  ~  P  is  perpendicular  to  QP  or  b,  and  hence,  from  the  prop- 
osition given  in  Sec.  60  to  63,  Qf  must  lie  in  a  line  through  Pf 
(which  coincides  with  P)  perpendicular  to  the  motion  of  Q  ~  P, 
that  is  in  a  line  through  P'  in  the  direction  of  b,  or  on  b  produced. 
Again,  the  direction  of  motion  of  Q  ~  R  is  perpendicular  to 
QR  or  c,  and  since  R'  at  0  has  the  same  motion  as  R,  both  being 
fixed,  this  is  also  the  direction  of  motion  of  Q  ~  R'}  so  that  Q'  lies 


56  THE  THEORY  OF  MACHINES 

on  a  line  through  Rr  perpendicular  to  the  motion  of  Q  ~  R,  that  is, 
on  the  line  through  R'  in  the  direction  of  c.  Now  as  Qf  has  been 
shown  to  lie  on  b  or  on  b  produced,  and  also  on  the  line  through  0 
parallel  to  c,  therefore  it  lies  at  the  point  shown  on  the  diagram 
at  the  intersection  of  these  two  lines. 

66.  Images  are  Points  on  the  Primary  Link. — It  may  be  well 
again  to  remind  the  reader  that  the  point  Qr  is  a  point  on  a  but 
that  its  motion  is  identical  with  that  of  Q  at  the  junction  of  the 
links  6  and  c.  If  the  angular  velocity  of  a  is  co  radians  per  second, 
then  the  linear  velocity  of  Qf  on  a  is  Q'O.u  ft.  per  second  and  its 
direction  in  space  is  perpendicular  to  Q'O,  and  from  the  sense  of 
rotation  shown  on  Fig.  32  it  moves  to  the  left.  Since  the  motion 
of  Q'  is  the  same  as  that  of  Q  then  Q  also  moves  to  the  left  in 
the  direction  normal  to  Q'O  and  with  the  velocity  Q'O.co  ft.  per 
second. 


67.  Images  of  Links. — Since  P'  and  Qf  are  the  images  of  P  and 
Q  on  b,  P'Q'  may  be  regarded  as  the  image  of  b,  and  will  in  future 
be  denoted  by  &';  similarly  R'Q'(OQ')  will  be  denoted  by  d '.  By 
a  similar  process  of  reasoning  it  may  be  shown  that  since  S  bisects 
PQ,  so  will  S'  bisect  P'Q'  and  also  Tr  may  be  found  from  the 
relation  R'T'  :  T'Q'  =  RT  :  TQ. 

Since  the  latter  point  is  of  importance  and  of  frequent  occur- 
rence, it  may  be  well  to  prove  the  method  of  locating  S'.  The 
direction  of  motion  of  S  ~  P  is  clearly  the  same  as  that  of  Q  ~  P, 
that  is,  perpendicular  to  PQ  or  b,  but  the  linear  velocity  of  Q  ~ 
P  is  twice  that  of  S  ~  P,  both  being  on  the  same  link  and  $ 
bisecting  PQ.  But  the  motion  of  P'  is  the  same  as  P  and  of  Q'  is 
the  same  as  Q;  hence  the  motion  of  Q'  ~  P'  is  exactly  the  same  as 
that  of  Q~P,  so  that  the  velocity  of  S'  ~  P'  is  one-half  that 


THE  MOTION  DIAGRAM  57 

of  Q'  ~  P',  or  S'  will  lie  on  PfQf  and  in  the  center  of  the  latter 
V)  line. 

68.  Angular  Velocities. — The  diagram  may  be  put  to  further 
use  in  determining  the  angular  velocities  of  b  and  c  when  that  of 
a  is  known  or  the  relation  between  them  when  that  of  a  is  not 
known.  Let  co&  and  o>c,  respectively,  denote  the  angular  velocities 
of  6  and  c  in  space,  the  angular  velocity  of  the  primary  link  a 
being  co  radians  per  second.  Now  Q  and  P  are  on  one  link  b  and 
the  motion  of  Q  ~  P  is  perpendicular  to  QP,  and  hence  the 
velocity  of  Q  ~  P  is  QP.ub  =  6.co6  ft.  per  second  where  o>&  is  the 
angular  velocity  of  6,  and  co&  is  as  yet  unknown.  Again  Qr  and 
P'  are  points  on  the  same  link  a,  which  turns  with  the  known 
angular  velocity  co,  and  hence  the  velocity  of  Q'  ~  P'  is  Q'P'.co  = 
b' co  ft.  per  second.  But  from  the  nature  of  the  case,  since  Q'  has 
the  same  motion  as  Q,  and  P'  the  same  motion  as  P,  the  velocity 
of  Q  ~  P  is  equal  to  that  of  Q'  ~  P'}  that  is,  6w&  =  6'w  or 


Similarly 


69.  Image   of  Link  Represents  Its  Angular  Velocity.  —  The 

above  discussion  shows  that  if  the  angular  velocity  of  a  is  con- 
stant then  the  lengths  of  the  images  b'  and  c'  represent  the  angular 

velocities  of  the  links  b  and  c  to  the  scale  JT  and  -  respectively, 

(j  C 

since  b  and  c  are  the  same  for  all  positions  of  the  mechanism.   On 

the  other  hand,  even  though  o>  is  variable,  at  any  instant  —  =  v' 

co         o 

etc.,  so  that  there  is  a  direct  method  of  getting  the  relation 
between  the  angular  velocities  in  such  cases. 

70.  Sense  of  Rotation  of  Links.  —  The  diagram  further  shows 
the  sense  in  which  the  various  links  are  turning,  and  by  the 
formulas  for  the  angular  velocities  these  are  readily  inferred. 

Thus  ub  =  -TV,  and  starting  at  the  point  P,  P'Q'  =  b'  is  drawn  to 

the  left  and  PQ  =  b  to  the  right,  hence  the  ratio  -j-  is  negative, 

or  the  link  b  is  at  the  instant  turning  in  opposite  sense  to  a  or 
in  a  clockwise  sense.     In  the  case  of  the  link  c  the  lines  R'Q'  and 


58 


THE  THEORY  OF  MACHINES 


RQ  are  drawn  upward  from  R  and  Rf,  that  is  —  is  positive  and 

c 

hence  a  and  c  are  turning  in  the  same  sense. 

71.  Phorograph  a  Vector  Diagram. — The  figure  OP'Q'R'  is 
evidently  a  vector  diagram  for  the  mechanism,  the  distance  of 
any  point  on  this  diagram  from  the  pole  0  being  a  measure  of  the 
velocity  of  the  corresponding  point  in  the  mechanism.  The 
direction  of  the  motion  in  space  is  normal  to  the  line  joining  the 
image  of  the  point  to  0,  and  the  sense  of  the  motion  is  known 
from  the  sense  of  rotation  of  the  primary  link.  Further,  the 
lengths  of  the  sides  of  this  vector  diagram,  b'(P'Q'),  c'(R'Q') 
and  d'(OR')  are  measures  of  the  angular  velocities  of  these  links 
the  sense  of  motion  being  determined  as  explained.  As  d  is  at 
rest,  OR'  has  no  length. 

In  Fig.  33  other  positions  and  proportions  of  a  similar  mechan- 
ism are  shown,  in  which  the  solution  is  given  and  the  results  will 
be. as  follows: 


FIG.  33. 

At  (1)  the  ratio  -r  is  positive  as  is  also  —  or  all  links  are  turning 
in  the  same  sense;  at  (2)  the  link  a  is  parallel  to  c  and  hence  Q1 
andP1  lie  at  P,  so  that  6'  =  0  or  w&  =  -rco  =  0,  that  is,  at  this  in- 
stant the  link  b  has  no  angular  velocity  and  is  either  at  rest  or 
has  a  motion  of  translation.  Evidently  it  is  not  at  rest  since  the 
velocity  of  all  points  on  it  are  not  zero  but  are  OP.co  =  ao>  ft. 
per  second.  As  shown  at  (3)  the  links  a  and  b  are  in  one  straight 
line  and  in  the  phorograph  Q'  and  Rf  both  lie  at  0,  so  that  Q'R'  = 
c'  =  0,  and  hence  ue  =  0,  in  which  case  the  link  c  is  for  the 
instant  at  rest,  since  both  Q'  and  R'  are  at  0,  the  only  point  at 

b'          c' 

rest  in  the  figure.     At  (4)  both  the  ratios  -r  and  —  are  negative 

o  c 

so  that  6  and  c  both  turn  in  opposite  sense  to  a  and  therefore 
in  the  same  sense  as  one  another.  At  (5)  the  parallelogram 
used  commonly  on  the  side  rods  of  locomotives,  is  shown  and  the 


THE  MOTION  DIAGRAM  59. 

phorograph  shows  that  Q'  and  P'  coincide  with  P  so  that  6'  =  0 
or  the  side  rod  b  has  a  motion  of  translation,  as  is  well  known. 
There  is  the  further  well-known  conclusion  that  since  c1  =  c  the 
links  a  and  c  turn  with  the  same  angular  velocity. 

It  is  to  be  noted  that  if  the  image  of  any  link  reduces  to  a 
single  point  two  explanations  are  possible:  (a)  if  this  point  falls 
at  0  the  link  is  stationary  for  the  instant  as  for  d  in  the  former 
figure,  and  also  as  for  c  in  (3)  of  Fig.  33 ;  but  (6)  if  the  point  is 
not  at  0,  the  inference  is  that  all  points  in  the  link  move  in  the 
same  way  or  the  link  has  a  motion  of  translation  at  the  instant, 
as  for  the  link  b  in  (2). 

The  method  will  now  be  employed  in  a  few  typical  cases. 

72.  Further  Example.— The  mechanism  shown  in  Fig.  34  is  a 
little  more  complicated  than  the  .previous  ones.  Here  P',  Qf 


FIG.  34. 

and  Rf  are  found  as  before,  and  since  the  motions  oiS^Q  and  S  >  —  >P 
are  perpendicular  respectively  to  SQ  and  SP,  therefore  S'P' 
and  S'Q'  are  drawn  parallel  respectively  to  SP  and  SQ,  thus 
locating  S'.  Also  T'  is  located  from  the  relation  R'T'  :  T'Q'  = 
RT  :  TQ  (Sec.  67).  Next  since  the  motions  of  U~S&ud  U  ~  Tare 
given,  draw  S'U'  parallel  to  SU  and  U'T'  parallel  to  UT,  their 
intersection  locating  Uf.  Assuming  a  to  turn  at  angular  veloc- 
ity co  in  the  sense  shown,  then  the  angular  velocity  of  SU  is 

Cf'  JJt  TT'  rfl' 

w  in  the  same  sense  as  a  and  that  of  UT  is  ~rjm  w  in  opposite 


sense  to  a  (Sec.  70).     The  linear  velocity  of  U  is  Ot/'.co  ft.  per 
second. 

73.  Image  is  Exact  Copy  of  Link.  —  There  is  an  important 
point  which  should  be  emphasized  here  and  it  is  illustrated  in 
finding  the  image  of  the  link  b.  The  method  shows  that  the  rela- 
tion, of  S'  to  P'Q'  is  the  same  as  that  of  S  to  PQ,  or  the  image 


60 


THE  THEORY  OF  MACHINES 


of  the  link  is  an  exact  copy  of  the  link  itself,  and  although  it  may 
be  inverted  and  is  usually  of  different  size,  all  lines  on  the  image 
are  parallel  with  the  corresponding  lines  on  the  mechanism. 
Whenever  the  image  of  the  link  is  inverted  it  simply  means  that 
the  link,  of  which  this  is  the  image,  is  turning  at  the  given  in- 
stant in  the  opposite  sense  to  the  link  of  reference;  if  the  image  is 
the  same  size  as  the  original  link,  then  the  link  has  the  same 
angular  velocity  as  the  link  reference. 

74.  Valve  Gear. — The  mechanism  shown  in  Fig.  35  is  very 
commonly  used  by  some  engine  builders  for  operating  the  slide 
valve,  OP  being  the  eccentric,  RS  the  rocker  arm  pivoted  to 
the  frame  at  R,  ST  the  valve  rod  and  T  the  end  of  the  valve 
stem  which  has  a  motion  of  sliding.  OP  has  been  selected  as 
the  link  of  reference  and  P',  Q',  R'  and  S'  are  found  as  before. 


FIG.  35. — Valve  gear. 

The  construction  forces  T  to  move  horizontally  in  space,  and 
therefore  T'  must  lie  on  a  line  through  0  perpendicular  to  the 
motion  of  T,  that  is  T'  is  on  a  vertical  line  through  0,  and  further 
T'  lies  on  a  line  through  S'  parallel  to  ST,  which  fixes  T'. 

Following  the  instructions  given  regarding  former  cases,  it  is 
evident  that  the  velocity  of  the  valve  is  OT'.u  ft.  per  second, 
co  being  the  angular  velocity  of  OP.  While  the  other  velocities 
are  not  of  much  importance  yet  the  figure  gives  the  angular 

S'T' 
velocity  of  ST  as  ~om~'w  in  the   opposite  sense  to  a  and  the 

linear  velocity  of  S  is  greater  than  that  of  T  in  the  ratio  OS':OT'. 

75.  Steam  Engine. — The  steam  engine  mechanism  is  shown  in 

Fig.  36,  (a)  where  the  piston  direction  passes  through  0  and  (6) 

where  it  passes  above  0  with  the  cylinder  offset,  but  the  same 


THE  MOTION  DIAGRAM 


61 


letters  and  description  will  apply  to  both.  Evidently  Qf  lies 
on  P'Q'  through  P',  parallel  to  PQ,  that  is,  on  QP  produced",  and 
also  since  the  motion  of  Q  in  space  is  horizontal,  Qf  will  lie  on  the 
vertical  through  0. 

The  velocity  of  the  piston  is  OQ'.co  in  both  cases  and  the  angular 

velocity  of  the  connecting  rod  is  ^ -co  in  the  opposite  sense  to  that 

of  the  crank,  since  P'Q'  lies  to  the  left  of  P'  while  PQ  lies  to  the 
right  of  P,  and  it  is  interesting  to  note  that  in  both  cases  when 
the  crank  is  to  the  left  of  the  vertical  line  through  0,  the  crank 
and  rod  turn  in  the  same  sense;  further  that  the  rod  is  not  turn- 


FIG.  36. 


ing  when  the  crank  is  vertical  because  Q'  and  P'  coincide  and 
hence  b'  =  0.  Again,  the  piston  velocity  will  be  zero  when  Q' 
lies,  at  0,  which  will  occur  when  a  and  b  are  in  a  straight  line; 
the  maximum  piston  velocity  will  be  when  OQ'  is  greatest  and  this 
will  not  occur  for  the  same  crank  angle  in  the  two  cases  (a) 
and  (b). 

If  a  comparison  is  made  between  the  two  figures  of  Fig.  36 
it  will  be  clear  that  the  length  OQ'  in  the  upper  figure  is  greater 
than  the  corresponding  length  in  the  lower  one,  or  the  upper 
piston  is  moving  at  this  instant  at  a  higher  rate  than  the  lower 
one,  but  if  the  whole  revolution  be  examined  the  reverse  will  be 
true  of  other  crank  positions;  in  fact,  the  lower  construction  is 
frequently  used  as  a  quick-return  motion  (see  Fig.  27). 


62 


THE  THEORY  OF  MACHINES 


76.  Whitworth  Quick-return  Motion. — The  Whitworth  quick- 
return  motion,  already  described  and  shown  at  Fig.  11,  is  illus- 
trated at  Fig.  37.  There  are  four  links  a,  b,  d  and  e  and  two 
sliding  blocks,  c  and  /,  d  being  the  fixed  and  a  the  driving 
link  which  rotates  at  speed  co,  and  is  selected  as  the  primary 
link.  P'  and  Q'  are  found  by  inspection.  Further,  S'  lies  on  a 
vertical  line  through  0,  and  R'  on  a  line  through  Q'  parallel  to 
QR,  but  the  exact  positions  of  S'  and  R'  are  unknown. 


s' 


FIG.  37. — Whitworth  quick-return  motion. 

Now  P  is  a  point  on  both  a  and  c.  Choose  T  on  b  exactly 
below  P  on  a,  T  thus  having  a  different  position  on  b  for  each 
position  of  a.  As  all  links  have  plane  motion,  the  only  motion 
which  T  can  have  relative  to  P  is  one  of  sliding  in  the  direction 
of  &,  or  the  motion  of  T  ~  P  is  in  the  direction  of  b;  hence  T'  lies 
on  a  line  through  P'  perpendicular  to  b.  But  T  is  a  point  on  the 
link  b  and  hence  Tf  must  be  on  a  line  through  Q'  parallel  with  b; 
thus  T'  is  determined,  and  having  found  T'  it  is  very  easy  to  find 
Rl  by  dividing  Q'T'  externally  at  R'  so  that 

Q'R'  =  QR. 
Q'T'  ~~~~  QT 

The  dotted  lines  show  a  simple  geometrical  method  for  finding 
this  ratio,  and  it  is  always  well  to  look  for  some  such  method  as  it 
enormously  reduces  the  time  involved  in  the  problem. 

Now  since  S  moves  horizontally,  S'  will  be  located  on  the 
vertical  line  through  0  and  also  on  the  line  R'S'  through  R' 
parallel  to  RS.  This  return  motion  is  frequently  used  on  shapers 


THE  MOTION  DIAGRAM  63 

and  other  machines,  the  tool  holder  being  attached  to  the  block 
/,  so  that  the  tool  holder  is  moving  with  linear  velocity  OS'.u  and 

O'T?' 
the  angular  velocity  of  the  link  b  is  ~7yn~  '<*  in  the  same  sense  as  a. 

It  must  be  noticed  that  although  P  and  T  coincide,  their  images 
do  not,  for  T  has  a  sliding  motion  with  regard  to  P  at  the  rate 
P'T'.c>)  ft.  per  second,  and  hence  both  cannot  have  the  same 
velocity.  If  P'  and  T'  coincided 
then  P  and  T  would  have  the 
same  velocity.  The  velocity  dia- 
gram for  S  for  the  complete 
revolution  of  a  is  shown  in  Fig.  38. 

77.  Stephenson  Link  Motion. — 
The  Stephenson  link  motion 
shown  in  Fig.  39  involves  a 
slightly  different  method  of  at- 
tack. The  proportions  have  been 
considerably  distorted  to  avoid  F">'  %$££&*£*  '" 
confusion  of  lines.  The  primary 

link  is  the  crankshaft  containing  the  crank  C  and  the  eccen- 
trics E  and  F,  and  the  scale  here  will  be  altered  so  that 
OC'  =  2  X  OC,  OE'  and  OF'  being  similarly  treated.  The 
scale  will  then  be  OC'  =  OC.co  ft.  per  second  or  J^co  :  1. 

The  points  C",  E',  F',  H',  Dr  and  /'  are  readily  located.  Further 
choose  M  on  the  curved  link  AGB  directly  below  K  on  the  rocker 
arm  LDK  and  draw  lines  E'A'}  F'B',  H'G'  and  D'K',  of  unknown 
lengths  but  parallel  respectively  to  EA,  FB,  HG  and  DK.  It 
has  already  been  seen  (Sec.  73)  that  the  image  of  each  link  is 
similar  to  and  similarly  divided  to  the  link  itself  (it  is,  in  fact,  a 
photographic  image  of  it) ;  hence  the  link  AGB  must  have  an 
image  similar  to  it,  that  is,  the  (imaginary)  straight  lines  A'G'  and 
G'B'  must  be  parallel  to  AG  and  GB  and  the  triangle  A'G'B' 
must  be  similar  to  AGB.  But  the  lines  on  which  A',  Gf  and  B'  lie 
are  known;  hence  the  problem  is  simply  the  geometrical  one  of 
drawing  a  triangle  A'G'B'  similar  and  parallel  to  AGB  with  its 
vertices  on  three  known  lines.  The  reader  may  easily  invent  a 
geometrical  method  of  doing  this  with  very  little  effort,  the  pro- 
cess being  as  simple  as  the  one  shown  in  Fig.  37,  but  the  con- 
struction is  not  shown  because  the  figure  is  already  complicated. 

Having  now  located  the  points  Af,  G'  and  B'  the  curved  link 
A'G'B'  may  be  made  by  copying  AGB  on  an  enlarged  scale  and  on 


64 


THE  THEORY  OF  MACHINES 


it  the  point  M'  may  be  located  similarly  to  M  in  the  actual  link. 
For  the  purpose  of  illustrating  the  problem  the  image  of  the 
curved  link  has  been  drawn  in  on  the  figure  although  this  is  not 
at  all  necessary  in  locating  M'.  Since  K  slides  with  regard  to  M , 
then  K'M'  is  drawn  normal  to  the  curved  link  at  M'  which 
locates  Kr  and  then  L'  is  found  from  the  relation  LD  :  DK  = 
L'D'  :  D'K'. 

The  construction  shows  that  the  curved  link  is  turning  in  the 

A'Bf 

same  sense  as  the  crank  with  angular  velocity  M"~T17  'w  since 

A.D 

the  scale  is  such  that  OE'  =  20E.     Again,  the  valve  is  moving  to 


Valve 


Roach  Rod 


FIG.  39. — Stephenson  link  motion. 

the  right  at  the  velocity  represented  by  OZ/,  and  further  the 
velocity  of  sliding  of  the  block  in  the  curved  link  is  represented 
by  K'M',  both  of  these  on  the  same  scale  that  OCr  =  2.00 
represents  the  linear  velocity  of  the  crankpin.  The  method 
gives  a  very  direct  means  of  studying  the  whole  link  motion. 

78.  Reeves  Valve  Gear. — The  Corliss  valve  gear  used  on  the 
Reeves  engine  is  shown  diagrammatically  in  Fig  40,  there  being 
no  great  attempt  at  proportions.  Very  little  explanation  is 
necessary;  O,  R  and  S  are  fixed  in  space,  S  being  the  end  of  the 
rocking  valve  stem;  hence  Rr  and  S'  are  at  0,  and  the  link  OP  is 
driven  direct  from  the  crankshaft  through  the  eccentric  connec- 


THE  MOTION  DIAGRAM 


(65) 


FIG,  40, — Reeves  valve  gear. 


FIG.  41, 


66  THE  THEORY  OF  MACHINES 

tion.  The  point  Q  on  the  sliding  block  e  is  directly  over  a  mova- 
ble point  T  on  the  lever  /  which  lever  is  keyed  to  the  valve  stem. 
The  image  of  the  link  /  is  found  by  drawing  S'T'  parallel  to  ST  and 
Tf  is  located  by  drawing  Q'T'  perpendicular  to  S'T'  or  to  ST.  In 

S'Tf 
this  position  the  angular  velocity  of  the  valve  is    ^  times  that 

of  the  link  OP,  and  from  this  data  the  linear  velocity  of  the  valve 
face  is  readily  found 

79.  Joy  Valve  Gear. — This  chapter  will  be  concluded  by  one 
other  example  here,  although  the  method  has  been  used  a  good 
deal  throughout  the  book  and  other  examples  appear  later  on. 
The  example  chosen  is  the  Joy  valve  gear,  shown  in  Fig.  41,  this 
gear  having  been  largely  used  for  locomotives  and  reversing 
engines.     Referring  to  the  figure,  a  is  the  crank,  b  the  connecting 
rod,  c  the  crosshead,  e  or  RST,  /  and  g  or  SWV  are  links  connected 
as  shown.     The  link  g,  to  which  the  valve  stem  is  connected  at 
V,  is  pivoted  to  a  block  h,  which  ordinarily  slides  in  a  slotted  link 
fixed  in  position,  but  in  order  to  reverse  the  engine  the  slotted 
block  is  thrown  over  to  the  dotted  position. 

There  should  be  no  difficulty  in  solving  this  problem,  the  only 
point  causing  any  hesitation  being  in  drawing  OW,  which  should 
be  normal  to  the  direction  of  motion  of  the  block  h.  The  velocity 
of  the  valve  is  OF1  •  co  in  opposite  sense  to  P  and  of  such  a 
point  as  S  is  OS1  •  w  in  a  direction  at  right  angles  to  OS1. 

80.  Important  Principles. — It  may  be  well  to  call  attention  to 
certain  fundamental  points  connected  with  the  construction  dis- 
cussed.    In  the  first  place,  it  will  be  seen  that  the  method  is  a 
purely  vector  one  for  representing  velocities,  and  is  thus  quite 
analogous  to  the  methods  of  graphic  statics..    As  in  the  latter 
case  the  idea  seems  rather  hard  to  grasp  but  the  application  will 
be  found  quite  simple,  and  even  in  complicated  mechanisms 
there  is  little  difficulty.     Graphical  methods  for  dividing  up 
lines  and  determining  given  ratios  are  worth  the  time  spent  in 
devising  them. 

It  should  be  remembered  that  in  the  phorograph  the  image  of 
a  link  is  a  true  image  of  the  actual  link,  that  is,  it  is  exactly 
similar  to  it,  and  similarly  divided,  and  is  always  parallel  to  the 
link  but  may  be  inverted  relative  to  it.  The  image  may  be  the 
same  size  or  larger  or  smaller  than  the  link  depending  on  how 
fast  it  is  revolving;  it  is  in  fact  exactly  what  might  be  seen  by 
looking  through  a  lens  at  the  link.  If  this  statement  is  kept  in 


THE  MOTION  DIAGRAM  67 

mind,  it  will  greatly  aid  in  the  solution  of  problems  and  the 
understanding  of  the  method. 

QUESTIONS  ON  CHAPTER  IV 

1.  Prove  that  any  point  in  a  body  can  only  move  relatively  to  any  other 
point,  in  a  direction  perpendicular  to  the  line  joining  them. 

2.  Define  the  phorograph  and  state  the  principles  involved. 

3.  A  body  a  rotates  about  a  fixed  center  0;  show  that  all  points  in  it  have 
different  velocities,  either  in  magnitude,  sense  or  direction. 

4.  Show  that  the  phorograph  is  a  vector  diagram.     What  quantities 
may  be  determined  directly  by  vectors  from  it? 

6.  If  the  image  of  a  link  is  equal  in  length  to  the  link  and  in  the  same 
sense,  what  is  the  conclusion?     What  would  it  be  if  the  image  was  a  point? 
*/  6.  In  the  mechanism  (3)  of  Fig.  9,  let  d  turn  at  constant  speed,  as  in  the 
Gnome  motor;  find  the  phorograph  and  the  angular  velocity  of  the  rod  6. 

7.  In  an  engine  of  30  in.  stroke  the  connecting  rod  is  90  in.  long,  and  the 
engine  runs  at  90  revolutions  per  minute.     Find  the  magnitude  and  sense  of 
the  angular  velocity  of  the  rod  for  crank  angles  45°,  135°,  225°  and  315°. 

8.  Make  a  diagram  of  a  Walschaert  valve  gear  and  find  the  velocity  and 
direction  of  motion  of  the  valve  for  a  given  crank  position. 

9.  Plot  the  angular  velocity  of  the  jaw  and  the  linear  velocity  of  the 
center  G  of  the  crusher  given  in  Chapter  XV,  Fig.  168.    See  also  Fig.  95. 


CHAPTER  V 
TOOTHED  GEARING 

81.  Forms  of  Drives. — In  machinery  it  is  frequently  necessary 
to  transmit  power  from  one  shaft  to  another,  the  ratio  of  the 
angular  velocities  of  the  shafts  being  known,  and  in  very  many 
cases  this  ratio  is  constant;  thus  it  may  be  desired  to  transmit 
power  from  a  shaft  running  at  120  revolutions  per  minute  to 
another  running  at  200  revolutions  per  minute.  Various  methods 
are  possible,  for  example,  pulleys  of  proper  size  may  be  attached 
to  the  shafts  and  connected  by  a  belt,  or  sprocket  wheels  may  be 
used  and  connected  by  a  chain,  as  in  a  bicycle,  or  pulleys  may  be 
placed  on  the  shafts  and  the  faces  of  the  pulleys  pressed  together, 
so  that  the  friction  between  them  may  be  sufficient  to  transmit 
the  power,  a  drive  used  sometimes  in  trucks,  or,  again,  toothed 
wheels  called  gear  wheels  may  be  used  on  the  two  shafts,  as  in 
street  cars  and  in  most  automobiles. 

Any  of  these  methods  is  possible  in  a  few  cases,  but  usually 
the  location  of  the  shafts,  their  speeds,  etc.,  make  some  one  of 
the  methods  the  more  preferable.  If  the  shafts  are  far  apart,  a 
belt  and  pulleys  may  be  used,  but  as  the  drive  is  not  positive  the 
belt  may  slip,  and  thus  the  relative  speeds  may  change,  the  speed 
of  the  driven  wheel  often  being  5  per  cent,  lower  than  the  diam- 
eters of  the  pulleys  would  indicate.  Where  the  shafts  are  fairly 
close  together  a  belt  does  not  work  with  satisfaction,  and  then  a- 
chain  and  sprockets  are  sometimes  used  which  cannot  slip,  and 
hence  the  speed  ratio  required  may  be  maintained.  For  shafts 
which  are  still  closer  together  either  friction  gears  or  toothed 
gears  are  generally  used.  Thus  the  nature  of  the  drive  will 
depend  upon  various  circumstances,  one  of  the  most  important 
being  the  distance  apart  of  the  shafts  concerned  in  it,  another 
being  the  question  as  to  whether  the  velocity  is  to  be  accurately 
or  only  approximately  maintained,  and  another  being  the  power 
to  be  transmitted. 

.  82.  Spur  Gearing. — The  discussion  here  deals  only  with  drives 
of  the  class  which  use  toothed  gears,  these  being  generally  used 
between  shafts  which  must  turn  with  an  exact  velocity  ratio  which 

68 


TOOTHED  GEARING  69 

must  be  known  at  any  instant,  and  they  are  generally  used  when 
the  shafts  are  fairly  close  together.  It  will  be  convenient  to 
deal  first  with  parallel  shafts,  which  turn  in  opposite  senses,  the 
gear  wheels  connecting  which  are  called  spur  wheels,  the  larger 
one  being  commonly  known  as  the  gear,  and  the  smaller  one  as 
the  pinion.  Kinematically,  spur  gears  are  the  exact  equivalent 
of  a  pair  of  smooth  round  wheels  of  the  same  mean  diameter,  and 
which  are  pressed  together  so  as  to  drive  one  another  by  friction. 
Thus  if  two  shafts  15  in.  apart  are  to  rotate  at  100  revolutions  per 
minute  and  200  revolutions  per  minute,  respectively,  they  may 
be  connected  by  two  smooth  wheels  10  in  and  20  in.  in  diameter, 
one  on  each  shaft,  which  are  pressed  together  so  that  they  will  not 
slip,  or  by  a  pair  of  spur  wheels  of  the  same  mean  diameter,  both 
methods  producing  the  desired  results.  But  if  the  power  to  be 
transmitted  is  great  the  friction  wheels  are  inadmissible  on  ac- 
count of  the  great  pressure  between  them  necessary  to  prevent 
slipping.  If  slipping  occurs  the  velocity  ratio  is  variable,  and 
such  an  arrangement  would  be  of  no  value  in  such  a  drive  as  is 
used  on  a  street  car,  for  instance,  on  account  of  the  jerky  motion 
it  would  produce  in  the  latter. 

83.  Sizes  of  Gears. — In  order  to  begin  the  problem  in  the 
simplest  possible  way  consider  first  the  very  common  case  of  a 
pair  of  spur  gears  connecting  two  shafts  which  are  to  have  a  con- 
stant velocity  ratio.  This  is,  the  ratio  between  the  speeds  HI 
and  n2  is  to  be  constant  at  every  instant  that  the  shafts  are  re- 
volving. Let  I  be  the  distance  from  center  to  center  of  the  shafts. 
Then,  if  friction  wheels  were  used,  the  velocities  at  their  rims 
will  be  irdiUi  and  Trd2n2  in.  per  minute,  where  di  and  dz  are  the 
diameters  of  the  wheels  in  inches,  and  it  will  be  clear  that  the 
velocity  of  the  rim  of  each  will  be  the  same  since  there  is  to  be  no 
slipping. 

Therefore 


since  ....         

and  i    riUi  =  r2n2  where  ri  and  r2  are  the  radii. 

But  n  +  r2  =  r^^~. 

Hence  -  r2  +  r2  =  I 

or  r2  =  — ^p I  in. 

and  ri  =  —      -  •  I  in. 


70 


THE  THEORY  OF  MACHINES 


Now,  whatever  actual  shape  is  given  of  these  wheels,  the  motion 
of  the  shafts  must  be  the  same  as  if  two  smooth  wheels,  of  sizes 
as  determined  above,  rolled  together  without  slipping.  In 
other  words,  whatever  shape  the  wheels  actually  have,  the  re- 
sulting motion  must  be  equivalent  to  that  obtained  by  the  roll- 
ing together  of  two  cylinders  centered  on  the  shafts.  In  gear 
wheels  these  cylinders  are  called  pitch  cylinders,  and  their  pro- 
jections on  a  plane  normal  to  their  axes,  pitch  circles,  and  the 
circles  evidently,  touch  on  a  line  joining  their  centers,  which 
point  is  called  the  pitch  point. 

84.  Proper  Outlines  of  Bodies  in  Contact. — Let  a  small  part 
of  the  actual  outline  of  each  wheel  be  as  shown  in  the  hatched 
lines  of  Fig.  42,  the  projections  on  the  wheels  being  required  to 


FIG.  42. 

prevent  slipping  of  the  pitch  lines.     It  is  required  to  find  the 
necessary  shape  which  these  projections  must  have. 

Let  the  actual  outlines  of  the  two  wheels  touch  at  P  and  let 
P  be  joined  to  the  pitch  point  C;  it  has  been  already  explained 
that  there  must  be  no  slipping  of  the  circles  at  C.  Now  P  is  a 
point  in  both  wheels,  and  as  a  point  in  the  gear  6  it  moves  with 
regard  to  C  on  the  pinion  a  at  right  angles  to  PC,  while  as  a 
point  on  the  pinion  a  it  moves  with  regard  to  the  gear  6  also  at 
right  angles  to  PC.  Whether,  therefore,  P  is  considered  as  a 
point  on  a  or  b  its  motion  must  be  normal  to  the  line  joining  it  to 
C.  A  little  consideration  will  show  that,  in  order  that  this  con- 
dition may  be  fulfilled,  the  shape  of  both  wheels  at  P  must  be 
normal  to  PC. 


TOOTHED  GEARING  71 

In  order  to  see  this  more  clearly,  examine  the  case  shown  in 
the  lower  part  of  the  figure,  where  the  projections  are  not  normal 
to  PiC  at  the  point  PI  where  they  touch.  From  the  very  nature 
of  the  case  sliding  must  occur  at  PI,  and  where  two  bodies  slide 
on  one  another  the  direction  of  sliding  must  always  be  along  the 
common  tangent  to  their  surfaces  at  the  point  of  contact,  that 
is,  the  direction  of  sliding  must  in  this  case  be  PiPr.  But  PI 
is  the  point  of  contact  and  is  therefore  a  point  in  each  wheel, 
and  the  motion  of  the  two  wheels  must  be  the  same  as  if  the  two 
pitch  circles  rolled  together,'  having  contact  at  C.  Such  being 
the  case,  if  the  two  projections  shown  are  placed  on  the  wheels, 
the  direction  of  motion  at  their  point  of  contact  should  be 
perpendicular  to  PiC,  whereas  here  it  is  perpendicular  to  PiC". 
This  would  cause  slipping  at  C,  and  would  give  the  proper 
shape  for  pitch  circles  of  radii  AC'  and  BC',  which  would  corre- 
spond to  a  different  velocity  ratio.  Thus  Cf  should  lie  at  C  and 
PiP'  should  be  normal  to  PiC. 

Another  method  of  dealing  with  this  matter  is  by  means  of 
the  virtual  center.  Calling  the  frame  which  supports  the  bear- 
ings of  a  and  b,  the  link  d,  then  A  is  the  center  ad  and  B  is  bd 
while  the  pitgh  point  C  is  ab.  It  is  shown  at  Sec.  33  that  the 
motion  of  b  with  regard  to  a  at  the  given  instant  is  one  of  rota- 
tion about  the  center  ab  and  hence  the  motion  of  P  in  b  is  normal 
to  PC.  Where  the  two  wheels  are  in  contact  at  P  there  is  rela- 
tive sliding  perpendicular  to  PC,  that  is,  at  P  the  surfaces  must 
have  a  common  tangent  perpendicular  to  PC.  The  shape 
shown  at  PI  is  incorrect  because  from  Sec.  33  the  center  ab  must 
lie  in  a  normal  through  PI  and  also  on  ad  —  bd  from  the 
theorem  of  the  three  centers,  Sec.  39;  so  that  it  would  lie  at 

C'.     But  if  ab  were  at  C\  then  —  =  -T-~r,  which  does  not   give 


the  ratio  required. 

85.  Conditions  to  be  Fulfilled.  —  From  the  foregoing  the  follow- 
ing important  statements  follow  :  The  shapes  of  the  projections 
or  teeth  on  the  wheels  must  be  such  that  at  any  point  of  contact 
they  will  have  a  common  normal  passing  through  the  fixed  pitch 
point,  and  while  the  pitch  circles  roll  on  one  another  the  pro- 
jections or  teeth  will  have  a  sliding  motion.  These  projections 
on  gear  wheejs  are  called  teeth,  and  for  convenience  in  manu- 
facturing, all  the  teeth  on  each  gear  have  the  same  shape,  al- 
though this  is  not  at  all  necessary  to  the  motion.  The  teeth 


72 


THE  THEORY  OF  MACHINES 


on  the  pinion  are  not  the  same  shape  as  those  on  the  gear  with 
which  it  meshes. 

There  are  a  great  many  shapes  of  teeth,  which  will  satisfy 
the  necessary  condition  set  forth  in  the  previous  paragraph,  but 
by  far  the  most  common  of  these  are  the  cycloidal  and  the  in- 
volute teeth,  so  called  because  the  curves  forming  them  are 
cycloids  and  involutes  respectively. 

86.  Cycloidal  Teeth. — Select  two  circles  PC  and  P'C,  Fig.  43, 
and  suppose  these  to  be  mounted  on  fixed  shafts,  so  that  the 


FIG.  43.— Cycloidal  teeth. 

centers  A  and  B  of  the  pitch  circles,  and  the  centers  of  the  de- 
scribing circles  PC  and  P'C,  as  well  as  the  pitch  point  C,  all  lie  in 
the  same  straight  line,  which  means  that  the  four  circles  are 
tangent  at  C.  Now  place  a  pencil  at  P  on  the  circle  PC  and  let 
all  four  circles  run  in  contact  without  slipping,  that  is,  the  cir- 
cumferential velocity  of  all  circles  at  any  instant  is  the  same.  As 
the  motion  continues  P  approaches  the  pitch  circles  eC  and  fC, 
and  if  the  right-hand  wheel  is  extended  beyond  the  circle  fCh,  the 
pencil  at  P  will  describe  two  curves,  a  shorter  one  Pe  on  the 
wheel  eCg  and  a  longer  one  Pf  on  the  wheel  fCh,  the  points  e  and 
/  being  reached  when  P  reaches  the  point  C,  and  from  the  condi- 
tions of  motion  arc  PC  =  arc  eC  =  fC. 

Now  P  is  a  common  point  on  the  curves  Pe  and  Pf  and  also 
a  point  on  the  circle  PC,  which  has  the  common  point  C  with 
the  remaining  three  circles.  Hence  the  motion  of  P  with  regard 


TOOTHED  GEARING  73 

to  eCg  is  perpendicular  to  PC,  and  of  P  with  regard  to  fCh  is 
perpendicular  to  PC,  that  is,  the  tangents  to  Pe  and  Pf  at  P 
are  normal  to _PC^ or  the  two  curves  have  a  common  tangent, 
and  hence  a  common  normal  PC  at  their  point  of  contact,  and 
this  normal  will  pass  through  the  pitch  point  C.  Thus  Pe  and 
Pf  fulfil  the  necessary  conditions  for  the  shapes  of  gear  teeth. 
Evidently  the  points  of  contact  along  these  two  curves  lie  along 
PC,  since  both  curves  are  described  simultaneously  by  a  point 
which  always  remains  on  the  circle  PC.  Now  these  curves  are 
first  in  contact  at  P  and  the  point  of  contact  travels  down  the 
arc  PC  relative  to  the  paper  till  it  finally  reaches  C  where  the 
points,  P,  C,  e  and  /  coincide,  so  that  since  Pe  is  shorter  than  Pf, 
the  curve  Pe  slips  on  the  curve  Pf  through  the  distance  Pf  —  Pe 
during  the  motion  from  P  to  C. 

Below  C  the  pencil  at  P  would  simply  describe  the  same  curves 
over  again  only  reversed,  and  to  further  extend  these  curves,  a 
second  pencil  must  be  placed  at  P'  on  the  right-hand  circle  P'C, 
which  pencil  will,  in  moving  downward  from  C,  draw  the  curves 
P'g  and  P'h,  also  fulfilling  the  necessary  conditions,  the  points 
of  contact  lying  along  the  arc  CPr  and  the  amount  of  slipping 
being  P'g  -  P'h. 

Having  thus  described  the  four  curves  join  the  two  formed  on 
wheel  eCg,  that  is  gPr  and  Pe,  forming  the  curve  Peg'P'i  and  the 
two  curves  on  fCh  as  shown  at  PfhiP'i,  and  in  this  way  long  curves 
are  obtained  which  will  remain  in  contact  from  P  to  P' ,  the  point 
of  contact  moving  relative  to  the  paper,  down  the  arcs  PCPf, 
and  the  common  normal  at  the  point  of  contact  always  passing 
through  C.  The  total  relative  amount  of  slipping  is  Pfh\Pf\  — 
Peg'P'i.  If  now  twro  pieces  of  wood  are  cut  out,  one  having  its 
side  shaped  like  the  curve  PeP'i  and  pivoted  at  A,  while  the 
other  is  shaped  like  PfP'i  and  pivoted  at  B',  then  from  what  has 
been  said,  the  former  may  be  used  to  drive  the  latter,  and  the 
motion  will  be  the  same  as  that  produced  by  the  rolling  of  the 
two  pitch  circles  together;  hence  these  shapes  will  be  the  proper 
ones  for  the  profiles  of  gear  teeth. 

87.  Cycloidal  Curves. — The  curves  Pe,  Pf,  P'g  and  P'h,  which 
are  produced  by  the  rolling  of  one  circle  inside  or  outside  of 
another,  are  called  cycloidal  curves,  the  two  Pe  and  P'h  being 
known  as  hypocycloids,  since  they  are  formed  by  the  describing 
circle  rolling  inside  the  pitch  circle,  while  the  two  curves  Pf  and 
P'g  are  known  as  epicycloidal  curves,  as  they  lie  outside  the  pitch 


74 


THE  THEORY  OF  MACHINES 


circles.  Gears  having  these  curves  as  the  profiles  of  the  teeth 
are  said  to  have  cycloidal  teeth  (sometimes  erroneously  called 
epicycloidal  teeth),  a  form  which  is  in  very  cpmmon  use.  So 
far  only  one  side  of  the  tooth  has  been  drawn,  but  it  will  be 
evident  that  the  other  side  is  simply  obtained  by  making  a 
tracing  of  the  curve  PeP\  on  a  piece  of  tracing  cloth,  with  center 
A  also  marked,  then  by  turning  the  tracing  over  and  bringing 
the  point  A  on  the  tracing  to  the  original  center  A  on  the  draw- 


FIG.  44.— Cycloidal  teeth. 

ing,  the  other  side  of  the  tooth  on  the  wheel  eCg  may  be  pricked 
through  with  a  needle.  The  same  method  may  be  employed 
for  the  teeth  on  wheel  fCh. 

The  method  of  drawing  these  curves  on  the  drafting  board  is  not 
difficult,  and  may  be  described.  Let  C  ...  5  in  Fig.  44  repre- 
sent one  of  the  pitch  circles  and  the  smaller  circles  the  describing 
circle.  Choose  the  arc  C5  of  any  convenient  length  and  divide 
it  into  an  equal  number  of  parts  the  arcs  C-l,  1-2,  etc.,  each 
being  so  short  as  to  equal  in  length  the  corresponding  chords. 
Draw  radial  lines  from  A  as  shown,  and  locate  points  G 


TOOTHED  GEARING  75 

distance  from  the  pitch  circle  equal  to  the  radius  of  the  describ- 
ing circle,  and  from  these  points  draw  in  a  number  of  circles 
equal  in  size  to  the  describing  circle.  Now  lay  off  the  arc  IM  = 
arc  1C,  and  arc  2N  equal  the  arc  C2  or  twice  the  arc  Cl,  and  the 
arc  3R  equal  three  times  arc  Cl,  etc.,  in  this  way  finding  the 
points  C,  M,  N,  R,  S,  which  are  points  on  the  desired  epicycloidal 
curve.  Similarly  the  hypocycloidal  curve  below  the  pitch  circle 
may  be  drawn. 

88.  Size  of  Describing  Circle. — Nothing  has  so  far  been  said 
of  the  sizes  of  the  describing  circles,  and,  indeed,  it  is  evident 
that  any  size  of  describing  circle,  so  long  as  it  is  somewhat  smaller 
than  the  pitch  circle,  may  be  used,  and  will  produce  a  curve  ful- 


FIG.  45. 

filling  the  desired  conditions,  but  it  may  be  shown  that  when  the 
describing  circle  is  one-half  the  diameter  of  the  corresponding 
pitch  circle  the  hypocycloid  becomes  a  radial  line  in  the  pitch 
circle,  and  for  reasons  to  be  explained  later  this  is  undesirable. 
The  maximum  size,  of  the  describing  circle  is  for  this  reason 
limited  to  one-half  that  of  the  corresponding  pitch  circle  and 
whenlTset  of  gears  are  to  run  together,  the  describing  circle  for 
the  set  is  usually  half  the  size  of  a  gear  having  from  12  to  15  teeth. 
This  will  enable  any  two  wheels  of  the  set  to  work  properly 
together. 

The  proof  that  the  hypocycloid  is  a  radial  line  if  the  describing 
circle  is  half  the  pitch  circle,  may  be  given  as  follows :  Let  ABC, 
Fig.  45,  be  the  pitch  circle  and  DPC  the  describing  circle,  P  being 
the  pencil,  and  BP  the  line  described  by  P  as  P  and  B  approach 
C.  The  arc  BC  is  equal  to  the  arc  PC  by  construction,  and  hence 


76 


THE  THEORY  OF  MACHINES 


the  angle  PEC  at  the  center  E  of  DPC  is  twice  the  angle  BDC, 
because  the  radius  in  the  latter  case  is  twice  that  in  the  former. 
But  the  angles  BDC  and  PEC  are  both  in  the  smaller  circle, 
the  one  at  the  circumference  and  the  other  at  the  center,  and 
since  the  latter  is  double  the  former  they  must  stand  on  the  same 
arc  PC.  In  other  words  BP  is  a  radial  line  in  the  larger  circle 
since  DP  and  DB  must  coincide. 

89.  Teeth  of  Wheels. — In  the  actual  gear  the  tooth  profiles 
are  not  very  long,  but  are  limited  between  two  circles  concentric 


FIG.  46. 

with  the  pitch  circle  in  each  gear,  and  called  the  addendum  and 
root  circles  respectively,  for  the  tops  and  bottoms  of  the 
teeth,  the  distances  between  these  circles  and  the  pitch  circle 
being  quite  arbitrarily  chosen  by  the  manufacturers,  although  cer- 
tain proportions,  as  given  later,  have  been  generally  adopted. 
These  circles  are  shown  on  Fig.  46  and  they  limit  the  path  of 
contact  to  the  reversed  curve  PCPi  and  the  amount  of  slipping 
of  each  pair  of  teeth  to  PR  -  PD  +  P^E  -  P^F  =  PR  +  PiE  - 
(PD  +  PiF),  the  distances  being  measured  along  the  profiles  of 
the  teeth  in  all  cases.  Further,  since  the  common  normal  to  the 
teeth  always  passes  through  C,  then  the  direction  of  pressure 
between  a  given  pair  of  teeth  is  always  along  the  line  joining  their 
point  of  contact  to  C,  friction  being  neglected,  the  limiting  direc- 
tions of  this  line  of  pressure  thus  being  PC  and  PiC. 


TOOTHED  GEARING  77 

The  arc  PC  is  called  the  arc  of  approach,  being  the  locus  of 
the  points  of  contact  down  to  the  pitch  point  C,  while  the  arc 
CPi  is  the  arc  of  recess,  PI  being  the  last  point  of  contact.  Sim- 
ilarly, the  angles  DAC  and  CAE  are  called  the  angle  of  approach 
and  angle  of  recess,  respectively,  for  the  left-hand  gear.  The 
reversed  curve  PCPi  is  the  arc  of  contact  and  its  length  depends 
to  some  extent  on  the  size  of  the  describing  circles  among  other 
things,  being  longer  as  the  relative  size  of  the  .  describing  circle 
increases.  If  this  arc  of  contact  is  shorter  than  the  distance  be- 
tween the  centers  of  two  adjacent  teeth  on  the  one  gear,  then  only 
one  pair  of  teeth  can  be  in  contact  at  once  and  the  running  is 
uneven  and  unsatisfactory,  while  if  this  arc  is  just  equal  to  the 
distance  between  the  centers  of  a  given  pair  of  teeth  on  one  gear, 
or  the  circular  pitch,  as  it  is  called  (see  Fig.  52),  one  pair  of  teeth 
will  just  be  going  out  of  contact  as  the  second  pair  is  coming  in, 
which  will  also  cause  jarring.  It  is  usual  to  make  PC  PI  at  least 
1.5  times  the  pitch  of  the  teeth.  ~This  will,  of  course,  increase  the 
amount  of  slipping  of  the  teeth. 

With  the  usual  proportions  it  is  found  that  when  the  number 
of  teeth  in  a  wheel  is  less  than  12  the  teeth  are  not  well  shaped 
for  strength  of  wear,  and  hence,  although  they  will  fulfil  the 
kinematic  conditions,  they  are  not  to  be  recommended  in  practice. 

90.  Involute  Teeth.  —  The  second  and  perhaps  the  most  com- 
mon method  of  forming  the  curves  for  gear  teeth  is  by  means  of 
involute  curves.  Let  A  and  B,  Fi$  A  |7,  represent  the  axes  of  the 
gears,  the  pitch  circles  of  which  touch  at  C,  and  through  C  draw 
a  secant  DCE  at  any  angle  6  to  the  normal  to  AB,  and  with 
centers  A  and  B  draw  circles  to  touch  the  secant  in  D  and  E. 

Now  (Sec.  83)  —  =    ^n  =  T^>  so  ^na^  ^ne  new  circles  have  the 

71  2 


same  speed  ratios  as  the  original  pitch  circles.  If  then  a  string 
is  run  from  one  dotted  circle  to  the  other  and  used  as  a  belt 
between  these  dotted  or  base  circles  as  they  are  called;  the  proper 
speed  ratio  will  be  maintained  and  the  two  pitch  circles  will 
still  roll  upon  one  another  without  slipping,  having  contact  at  C. 
Now,  choose  any  point  P  on  the  belt  DE  and  attach  at  this 
point  a  pencil,  and  as  the  wheels  revolve  it  wilt  evidently  mark 
on  the  original  wheels  having  centers  at  A  and  B,  two  curves 
Pa  and  Pb  respectively,  a  being  reached  when  the  pencil  gets 
down  to  E,  and  b  being  the  starting  point  just  as  the  pencil  leaves 


78 


THE  THEORY  OF  MACHINES 


D,  and  since  the  point  P  traces  the  curves  simultaneously  they 
will  always  be  in  contact  relation  to  the  paper  at  some  point 
along  DE,  the  point  of  contact  traveling  downward  with  the 
pencil  at  P.  Since  P  can  only  have  a  motion  with  regard  to  the 


FIG.  47. — Involute  teeth. 

wheel  aE  normal  to  the  string  PE,  and  its  motion  with  regard 
to  the  wheel  Db  is  at  right  angles  to  PD,  it  will  be  at  once  evident 


FIG.  48. — Involute  teeth. 


that  these  two  curves  have  a  common  normal  at  the  point  where 
they  are  in  contact,  and  this  normal  evidently  passes  through  C. 
Hence  the  curves  may  be  used  as  the  profiles  of  gear  teeth 
(Sec,  85). 


TOOTHED  GEARING  79 

4^-u7-" 

The  method  of  describing  these  curves  on  the  drafting  board 
is  as  follows:  Draw  the  base  circle  6-5  with  center  B,  Fig.  48, 
and  lay  off  the  short  arcs  6-1,  1-2,  2-3,  3-4,  etc.,  all  of  equal 
length  and  so  short  that  the  arc  may  be  regarded  as  equal  in 
length  to  the  chord.  Draw  the  radial  lines  Bb,  B  -  1,  B  -2,  etc., 
and  the  tangents  bD,  1-E,  2-F,  3-G,  4-H  any  length,  and 
lay  off  4  —  H  =  arc  5  —  4,  3  —  G  =  arc  3  —  5  which  equals  twice  arc 
5-4,  2-F  equal  three  times  arc  5-4,  1-E  equal  four  times 
arc  5-4,  etc.;  then  D,  E,  F}  G,  H  and  5  are  all  points  on  the 
desired  curve  and  the  latter  may  now  be  drawn  in  and  extended, 
if  desired,  by  choosing  more  points  below  b. 

91.  Involute  Curves. — The  curves  Pa  and  Pb,  Fig.  47,  are  called 
involute  curves,  and  when  they  are  used  as  the  profiles  of  gear 
teeth  the  latter  are  involute  teeth.     The  angle  6  is  the  angle  of 
obliquity,  and  evidently  gives  the  direction  of  pressure  between 
the  teeth,  so  that  the  smaller  this  angle  becomes  the  less  will  be 
the  pressure  between  the  teeth  for  a  given  amount  of  power  trans- 
mitted.    If,  on  the  other  hand,  this  angle  is  unduly  small,  the 
base  circles  approach  so  nearly  to  the  pitch  circles  in  size  that  the 
curves  Pa  and  Pb  have  very  short  lengths  below  the  pitch  circles. 
Many  firms  adopt  for  0  the  angle  14J^°,  in  which  case  the  diameter 
of  the  base  circle  is  0.968  (about  3^2)  that  of  the  pitch  circle. 
If  the  teeth  are  to  be  extended  inside  the  base  circles,  as  is  usual, 
the  inner  part  is  made  radial.     With  teeth  of  this  form  the  dis- 
tance between  the  centers  A  and  B  may  be  somewhat  increased 
without  affecting  in  any  way  the  regularity  of  the  motion. 
Recently  some  makers  of  gears  for  automobile  work  have  in- 
creased the  angle  of  obliquity  to  20°,  in  this  way  making  the 
teeth  much  broader  and  stronger.     Stub  teeth  to  be  discussed 
later,  are  frequently  made  in  this  way,  largely  for  use  on  auto- 
mobiles.    A  discussion  of  the  forms  of  teeth  appears  in  a  later 
section. 

92.  Sets  of  Wheels  with  Involute  Teeth. — Gears  with  involute 
teeth  are  now  in  very  common  use,  and  if  a  set  of  these  is  to  be 
made,  any  two  of  which  are  capable  of  working  together,  then 
all  must  have  the  same  angle  of  obliquity.     The  arc  of  contact 
is  usually  about  twice  the  circular  pitch  and  the  number  of  teeth 
in  a  pinion  should  not  be  less  than  12  as  the  teeth  are  liable  to 
be  weak  at  the  root  unless  the  angle  of  obliquity  is  increased. 

A  more  complete  drawing  of  a  pair  of  gears  having  involute 
teeth  is  shown  in  Fig.  49.  Taking  the  upper  gear  as  the  driver, 


80  THE  THEORY  OF  MACHINES 

the  line  of  'Contact  will  be  along  DPCE,  but  the  addendum  circles 
usually  limit  the  length  of  this  contact  to  some  extent,  contact 
taking  place  only  on  the  part  of  the  obliquity  line  DE  inside  the 
addendum  line.  The  larger  the  addendum  circles  the  longer  the 
lines  of  contact  will  be  and  the  proportions  are  such  in  Fig  49 
that  contact  occurs  along  the  entire  line  DE.  No  contacts  can 
possibly  occur  inside  the  base  circles. 


nterference  Line 
Upper  Gear 


Interference  Line 
Lower  Gear 


FIG.  49. — Involute  teeth. 

93.  Racks. — When  the  radius  of  one  of  the  gears  becomes 
infinitely  large  the  pitch  line  of  it  becomes  a  straight  line  tangent 
to  the  pitch  line  of  the  other  gear  and  it  is  then  called  a  rack. 
The  teeth  of  the  rack  in  the  cycloidal  system  are  made  in  exactly 
the  same  manner  as  those  of  an  ordinary  gear,  but  both  the  de- 
scribing circles  roll  along  a  straight  pitch  line,  generating  cycloidal 
curves,  having  the  same  properties  as  those  on  the  ordinary 
gear. 

For  the  involute  system  the  teeth  on  the  rack  simply  have 
straight  sides  normal  to  the  angle  of  obliquity,  each  side  of  such 
teeth  forming  the  angle  6  with  the  radius  line  AC  drawn  from 
the  center  of  the  pinion  to  the  pitch  point. 

94.  Annular  or  Internal  Gears. — In  all  cases  already  discusesd 
the  pair  of  gears  working  together  have  been  assumed  to  turn  in 
opposite  sense,  resulting  in  the  use  of  spur  gears,  but  it  not  in- 
frequently happens  that  it  is  desired  to  have  the  two  turn  in  the 
same  sense,  in  which  case  the  larger  one  of  the  gears  must  have 
teeth  inside  the  rim  and  is  called  an  annular  or  internal  gear. 
An  annular  gear  meshes  with  a  spur  pinion,  and  it  will  be  self- 
evident  that  the  annular  gear  must  always  be  somewhat  larger 
than  the  pinion. 

A  small  part  of  annular  gears  both  on  the  cycloidal  and  in- 
volute systems  is  shown  at  Fig.  50  and  the  odd  appearance  of  the 


TOOTHED  GEARING 


81 


involute  internal  gear  teeth  is  evident;  such  gears  are  frequently 
avoided  by  the  use  of  an  extra  spur  gear. 


Cycloidal  Teeth 


Involute  Teeth 


FIG.  50. — Internal  gears. 

95.  Interference. — The  previous  discussion  deals  with  the  cor- 
rect theoretical  form  of  teeth  required  to  give  a  uniform  velocity 


FIG.  51. — Interference. 

ratio,  but  with   the  usual  proportions  adopted  in  practice  for 
the  addendum,  pitch  and  root  circles,  it  is  found  that  in  certain 


82  THE  THEORY  OF  MACHINES 

cases  parts  of  teeth  on  one  of  the  gears  would  cut  into  the  teeth 
on  the  other  gear,  causing  interference.  This  is  most  common 
with  the  involute  system  and  occurs  most  where  the  difference 
in  size  of  the  gears  in  contact  is  greatest;  thus  interference  is 
worst  where  a  small  pinion  and  a  rack  work  together,  but  it 
may  occur,  to  some  extent  with  all  sizes  of  gears. 

An  example  will  make  this  more  clear.  The  drawing  in  Fig.  51 
represents  one  of  the  smaller  pinions  geared  with  a  rack  in  the 
involute  system  and  it  is  readily  seen  that  the  point  of  the  rack 
tooth  cuts  into  the  root  of  the  tooth  on  the  gear  at  H  and  that  in 
order  that  the  pair  may  work  together  it  will  be  necessary  either 
to  cut  away  the  bottom  of  the  pinion  tooth  or  the  top  of  the 
rack  tooth.  This  conflict  between  the  two  sets  of  teeth  is  called 
interference. 

Looking  at  the  figure,  and  remembering  the 'former  discussion 
(Sec.  90)  on  involute  teeth,  it  is  seen  that  contact  will  be  along 
the  line  of  obliquity  from  C  to  E  and  that  points  on  this  line  CE 
produced  have  no  meaning  in  this  regard,  so  that  if  BC  denote 
the  pitch  line  of  the  rack,  the  teeth  of  the  rack  can  only  be  use- 
fully extended  up  to  the  line  ED,  whereas  the  actual  addendum 
line  is  FG.  Thus,  the  part  of  the  rack  teeth  between  ED  and  FG, 
as  shown  hatched  on  one  tooth  on  the  right,  cannot  be  made  the 
same  shape  as  the  involute  would  require  but  must  be  modified 
in  order  to  clear  the  teeth  of  the  pinion.  The  usual  practice 
is  to  modify  the  teeth  on  the  rack,  leaving  the  lower  parts  of  the 
teeth  on  the  pinion  unchanged,  and  the  figure  shows  dotted  how 
the  teeth  of  the  rack  are  trimmed  off  at  the  top  to  make  proper 
allowance  for  this. 

Interference  will  occur  where  the  point  E,  Fig.  51  (a),  falls  below 
the  addendum  line  FG,  the  one  tooth  cutting  into  the  other  at  H 
on  the  line  of  obliquity.  Where  a  pinion  meshes  with  a  gear  which 
is  not  too  large,  then  the  curvature  of  the  addendum  line  of  the 
gear  may  be  sufficient  to  prevent  contact  at  the  point  H,  in  which 
case  interference  will  not  occur.  As  has  already  been  explained, 
interference  occurs  most  when  a  pinion,  meshes  with  a  gear 
which  is  very  much  larger,  or  with  a  rack.  Where  a  large 
gear  meshes  with  a  rack  as  in  the  diagram  at  Fig.  51  (6),  the 
interference  line  DE  is  above  the  addendum  line  FG  and  hence 
no  modification  is  necessary. 

In  Fig.  49  the  interference  line  for  the  lower  gear  is  inside  the 
addendum  line  and  hence  the  points  of  these  teeth  must  be  cut 


TOOTHED  GEARING 


83 


away,  but  the  points  of  the  teeth  on  the  upper  gear  would  be 
correct  as  the  interference  line  for  it  coincides  with  its  adden- 
dum line. 

96.  Methods  of  Making  Gears. — Gear  wheels  are  made  in 
various  ways,  such  as  casting  from  a  solid  pattern,  or  from  a 
pattern  on  a  moulding  machine  containing  only  a  few  teeth, 
neither  of  which  processes  give  the  most  accurate  form  of  tooth. 
The  only  method  which  has  been  devised  of  making  the  teeth 
with  great  accuracy  is  by  cutting  them  from  the  solid  casting, 
and  the  present  discussion  deals  only  with  cut  teeth.  In  order 
to  produce  these,  a  casting  or  forging  is  first  accurately  turned 
to  the  outside  diameter^^lheKeeth,  that  is  to  the  diameter  of 
the  addendum  line,  and  the  metal  forming  the  spaces  between  the 
teeth  is  then  carefully  oBKmt  by  machine,  leaving  accurately 
formed  teeth  if  the  work  is  well  done.  Space  does  not  permit 
the  discussion  of  the  machinery  for  doing  this  class  of  work,  for 
various  principles  are  used  in  them  and  a  number  of  makes  of 
the  machines  will  produce  theoretically  correct  tooth  outlines. 
The  reader  will  be  able  to  secure  information  from  the  builders 
of  these  machines  himself. 


FIG.  52. 


-t 


97.  Parts  of  Teeth. — The  various  terms  applied  to  gear  teeth, 
either  of  the  involute  or  cycloidal  form,  will  appear  from  Fig. 
52.  The  addendum  line  is  the  circle  whose  diameter  is  that  of 
the  outside  of  the  gear,  the  dedendum  line  is  a  circle  indicating 
the  depth  to  which  the  tooth  on  the  other  gear  extends;  usually 
the  addendum  and  dedendum  lines  are  equidistant  from  the 
pitch  line.  The  teeth  usually  are  cut  away  to  the  root  circle 
which  is  slighly  inside  the  dedendum  circle  to  allow  for  some 
clearance,  so  that  the  total  depth  of  the  teeth  somewhat  exceeds 


84  THE  THEORY  OF  MACHINES 

the  working  depth  or  distance  between  the  addendum  and  de- 
dendum  circles.  The  dimension  or  length  of  the  tooth  parallel 
to  the  shaft  is  the  width  of  face  of  the  gear,  or  often  only  the 
face  of  the  gear,  while  the  face  of  the  tooth  is  the  surface  of  the 
latter  above  the  pitch  line  and  the  flank  of  the  tooth  is  the  surface 
of  the  tooth  below  the  pitch  line.  The  solid  part  of  the  tooth 
outside  the  pitch  line  is  the  point  and  the  solid  part  below  the 
pitch  line  is  the  root. 

Two  systems  of  designating  cut  teeth  are  now  in  use,  the  one 
most  commonly  used  being  by  Brown  and  Sharpe  and  it  will 
first  be  described. 

Let  d  be  the  pitch  diameter  of  a  gear  having  t  teeth,  h\  the  depth 
of  the  tooth  between  pitch  and  addendum  circles,  and  /i2  the  depth 
below  the  pitch  circle,  so  that  the  whole  depth  of  the  tooth  is 
h  =  hi  +  h2,  while  the  working  depth  is  2hi.  The  distance 
measured  along  the  circumference  of  the  pitch  circle  from  center 
to  center  of  teeth  is  called  the  circumferential  or  circular  pitch 
which  is  denoted  by  p  and  it  is  evident  that  pt  =  ird.  In  the  case 
of  cut  teeth  the  width  W  of  the  tooth  and  also  of  the  space  along 
the  pitch  circle  are  equal,  that  is,  the  width  of  the  tooth  measured 
around  the  circumference  of  the  pitch  circle  is  equal  to  one-half 
the  circular  pitch.  The  statements  in  the  present  paragraph  are 
true  for  all  systems. 

98.  System  of  Teeth  Used  by  Brown  and  Sharpe.  —  Brown  and 
Sharpe  have  used  very  largely  the  term  diametral  pitch  which  is 
defined  as  the  number  of  teeth  divided  by  the  diameter  in  inches 
of  the  pitch  circle,  and  the  diametral  pitches  have  been  largely 
confined  to  whole  numbers  though  some  fractional  numbers  have 
been  introduced.  Thus  a  gear  of  5  diametral  pitch  means  one 
in  which  the  number  of  teeth  is  five  times  the  pitch  diameter  in 
inches,  that  is  such  a  gear  having  a  pitch  diameter  of  4  in.  would 

have  20  teeth.     Denoting  the  diametral  pitch  by  q  then  q  =  - 

and  from  this  it  follows  that  pq  =  TT  or  the  product  of  the  diame- 
tral and  circular  pitches  is  3.1416  always.  The  circular  pitch 
is  a  number  of  inches,  the  diametral  pitch  is  not. 

The  standard  angle  of  obliquity  used  by  Brown  and  Sharpe 

is  14J^°  and  further  hi=  —  in.,/i2  =  —    +  ^  in.,  clearance  =  —^ 


in.,  and  the  width  W  of  the  tooth  is  £,  so  that  there  is  no  side 

A 

clearance  or  back  lash  between  the  sides  of  the  teeth. 


TOOTHED  GEARING  85 

99.  Stub   Tooth   System. — Recently   the   very   great   use   of 
gears  for  automobiles  and  the  severe  service  to  which  these 
gears  have  been  put  has  caused  manufacturers  to  introduce  what 
is  often  called  the  "  Stub  Tooth  "  system  in  which  the  teeth  are 
not  proportioned  as  adopted  by  Brown  and  Sharpe.     Stub  teeth 
are  made  on  the  involute  system  with  an  obliquity  of  20°  usually, 
and  are  not  cut  as  deep  as  the  teeth  already  described.     The  di- 
mensions of  the  teeth  are  designated  by  a  fraction,  the  numerator 
of  which  indicates  the  diametral  pitch  used,  while  the  denomina- 
tor shows  the  depth  of  tooth  above  the  pitch  line.     A  %  gear 
is  one  of  5  diametral  pitch  and  having  a  tooth  of  depth  hi  = 
J<7  in.  above  the  pitch  line  (in  the  Brown  and  Sharpe  system 
Tii  would  be  J^  in.  for  the  same  gear). 

The  usual  pitches  with  stub  tooth  gears  are  ££,  j^f,  %,  %,  %Q, 
%l>  1?f  2  and  1M4-  Some  little  difference  of  opinion  appears 
to  exist  with  regard  to  the  clearance  between  the  tops  and  roots 
of  the  teeth,  the  Fellows  Gear  Shaper  Co.  making  the  clear- 
ance equal  to  one-quarter  of  the  depth  hi.  Thus,  a  %  gear 
would  have  the  same  hi  as  is  used  in  the  Brown  and  Sharpe 
system  for  a  7  diametral  pitch  gear,  that  is  hi  =  0.1429  in.,  and 
a  clearance  equal  to  0.25  X  J<f  =  0.0357  in.,  which  is  much 
greater  than  the  0.0224  in.  which  would  be  used  in  the  Brown 
and  Sharpe  system  on  a  7  pitch  gear. 

100.  The  Module. — In  addition  to  the  methods  already  ex- 
plained of  indicating  the  size  of  gear  teeth,  by  means  of  the  cir- 
cular and  diametral  pitch,  the  module  has  also  to  some  extent 
been  adopted,  more  especially  where  the  metric  system  of  meas- 
urement is  in  use.     The  module  is  the  number  of  inches  of 
diameter  per  tooth,  and  thus  corresponds  with  the  circular  pitch, 
or  number  of  inches  of  circumferences  per  tooth,  and  is  clearly 
the  reciprocal  of  the  diametral  pitch.     Using  the  symbol  m  for 
the  module  the  three  numbers  indicating  the  pitch  are  related 
as  follows : 

1  1        TT 

m  =  -;  also  a  —  —  =  -• 

q'  m       p 

.    The  module  is  rarely  expressed  in  other  units  than  millimeters. 

101.  Examples. — A  few  illustrations  will  make  the  use  of  the 
formulas  clear,  and  before  working  these  it  is  necessary  to  re- 
member that  any  pair  of  gears  working  together  must  have  the 
same  pitch  and  a  set  of  wheels  constructed  so  that  any  two  may 


86  THE  THEORY  OF  MACHINES 

work  together  must  have  the  same  pitch  and  be  designed  on  the 
same  system. 

Let  di  and  dz  be  the  pitch  diameters  of  two  gears  of  radii 
ri  and  r2  respectively  and  let  these  be  placed  on  shafts  I  in.  apart 
and  turning  at  HI  and  n2  revolutions  per  minute.  Then  from 
Sec.  83,  where  spur  gears  only  are  used, 


and 
also 


=  __ 

HI  -h  nz 


Suppose  I  =  9  in.  between  centers  of  shafts  which  turn  at 
100  revolutions  and  200  revolutions  per  minute;  then,  substitut- 
ing in  the  above  formula  n  =  inn  7-57^  X  9  =  6  in.  and  r2  = 

J.UU  "T~  ^UU 

100 

100  -H200  *  9  =  3  m'J  or  tlie  gears  w^  ke  1-  m>  anc*  ^Jn'  ^" 
ameter  respectively.  If  cut  to  4  diametral  pitch  the  numbers  of 
teeth  will  be  ti  =  4  X  12  =  48  and  t2  =  4  X  6  =  24.  The  cir- 
cular pitch  is  4  =  0.7854  in.  Further,  hi  =  Y±  in.  and  the 
outside  diameters  of  the  gears  are  12 J^  in.  and  6%  in.,  the  tooth 
clearances  =  ~~^ —  =  0.0393  in.  The  nodule  would  be  J£  in. 

6  in. 
~  24  teeth* 

If  the  gears  have  stub  teeth  of  four-fifths  size,  then  the  numbers 
of  teeth  will  be  48  and  24  as  before,  but  hi  will  be  %  =  0.2  in.,  so 
that  the  outside  diameters  will  be  12.4  in.  and  6.4  in.  respectively, 
the  clearance  will  be  J4  X  J4  =  0.0625  in.  and  the  total  depth 
of  the  teeth  0.4625  in.  as  compared  with  0.5393  in.  for  the  teeth 
on  the  Brown  and  Sharpe  system. 

Inasmuch  as  it  is  rather  more  usual  to  use  the  outside  diameter 
of  a  gear  than  the  pitch  diameter  in  shops  where  they  are  made, 
it  is  very  desirable  that  the  reader  become  so  familiar  with  the 
proportions  as  to  be  able  to  know  instantly  the  relations  between 
the  different  dimensions  of  the  gears  in  terms  of  the  outside  di- 
ameter aud  the  pitch. 


TOOTHED  GEARING  87 

102.  Discussion  of  the  Gear  Systems. — The  involute  form  of 
tooth  is  now  more  generally  used  than  the  cycloidal  form.     In  the 
first  place  the  profile  is  a  single  curve  instead  of  the  double  one 
required  with  the  cycloidal  shape.    Again,  because  of  its  construc- 
tion, it  is  possible  to  separate  the  centers  of  involute  gears  without 
causing  any  unevenness  of  running,  that  is,  if  the  gears  are  de- 
signed for  shafts  at  certain  distance  apart  this  distance  may  be 
slightly  increased  without  in  the  least  altering  the  velocity  ratio 
or  disturbing  the  evenness  of  the  running,  this  is  an  advantage 
not  possessed  by  cycloidal  teeth. 

In  cycloidal  teeth  the  direction  of  pressure  between  a  given 
pair  of  teeth  is  variable,  being  always  along  the  line  joining  the 
pitch  point  to  the  point  of  contact,  and  when  the  point  of  con- 
tact is  the  pitch  point,  the  direction  of  pressure  is  normal  to  the 
line  joining  the  shaft  centers.  In  the  involute  teeth  the  pressure 
is  always  in  the  same  direction  being  along  the  line  of  obliquity, 
and  thus  the  pressure  between  the  teeth  and  the  force  tending 
to  separate  the  shafts  is  somewhat  greater  in  the  involute  form, 
although  there  is  no  very  great  advantage  in  cycloidal  teeth 
from  this  point  of  view.  The  statements  in  this  paragraph  as- 
sume that  there  is  no  friction  between  the  teeth. 

Interference  is  somewhat  greater  in  involute  teeth. 

As  regards  the  Brown  and  Sharpe  proportions  and  the  stub 
teeth,  of  course  the  large  angle  of  obliquity  of  the  latter  teeth 
increases  the  pressure  for  the  power  transmitted.  The  stub  tooth 
gears  are,  however,  stronger  and  there  is  very  little  interference 
owing  to  the  shorter  tooth.  The  teeth  would  possibly  be  a 
little  cheaper  to  cut,  and  this  as  well  as  their  greater  strength 
would  give  them  a  considerable  advantage  in  such  machines  as 
automobiles. 

103.  Helical  Teeth. — A  study  of  such  drawings  as  are  shown  at 
Fig.  46,  etc.,  shows  that  the  smaller  the  depth  of  the  teeth  the 
less  will  be  the  amount  of  slipping  and  therefore  the  less  the 
frictional  loss.     But  this  is  also  accompanied  by  a  decrease  in  the 
arc  of  contact  and  hence  the  number  of  teeth  in  contact  at  any 
one  time  will,  for  a  given  pitch,  be  decreased,  which  may  cause 
unevenness  in  the  motion.     If,  however,  the  whole  width  of  the 
gear  be  assumed  made  up  of  a  lot  of  thin  discs,  and  if,  after  the 
teeth  had  been  cut  across  all  the  discs  at  once,  they  were  then 
slightly  twisted  relatively  to  one  another,  then  the  whole  width 
of  the  gear  would  be  made  up  of  a  series  of  steps  and  if  these 


88  THE  THEORY  OF  MACHINES 

steps  were  made  small  enough  the  teeth  would  run  across  the 
face  of  the  gear  as  helices,  and  the  gear  so  made  would  be  called 
a  helical  gear.  The  advantage  of  such  gears  will  appear  very 
easily,  for  instead  of  contact  taking  place  across  the  entire  width 
of  the  face  of  a  tooth  at  one  instant,  the  tooth  will  only  gradually 
come  into  contact,  the  action  beginning  at  one  end  and  working 
gradually  over  to  the  other,  and  in  this  way  very  great  evenness 
of  motion  results,  even  with  short  teeth  of  considerable  pitch. 

The  profile  of  the  teeth  of  such  gears  is  made  the  same,  on  a 
plane  normal  to  the  shaft,  as  it  would  be  if  they  were  ordinary 
spur  gears. 


FIG.  53. — Double  helical  gears. 

Helical  gears  are  a  necessity  in  any  case  where  high  speed  and 
velocity  ratio  are  desired,  and  the  modern  reduction  gear  now 
being  much  used  between  steam  turbines  and  turbine  pumps  and 
dynamos  would  be  a  failure,  on  account  of  the  noise  and  vibra- 
tion, if  ordinary  spur  gears  were  used.  Such  reduction  gears  are 
always  helical  and  frequently  two  are  used  with  the  teeth  run- 
ning across  in  opposite  directions  so  as  to  avoid  end  thrust. 
Some  turbines  have  been  made  in  which  such  gears,  running  at 
speeds  of  over  400  revolutions  per  second,  have  worked  without 
great  noise. 

A  photograph  of  a  De  Laval  double  helical  gear  is  shown  in 
Fig.  53,  this  gear  being  used  to  transmit  over  1,000  hp.  without 
serious  noise.  In  the  figure  the  teeth  run  across  the  face  of  the 
gear  at  about  45°,  and  are  arranged  to  run  in  oil  so  as  to  prevent 
undue  friction. 


TOOTHED  GEARING  89 

QUESTIONS  ON  CHAPTER  V 

A  shaft  running  at  320  revolutions  per  minute  is  to  drive  a  second  one 
20  in.  away  at  80  revolutions  per  minute,  by  means  of  spur  gears;  find  their 
pitch  diameters. 

Xf2.  If  both  shafts  were  to  turn  in  the  same  sense  at  the  speeds  given  and 
were  4  in.  apart,  what  would  be  the  sizes  of  the  gears? 

3.  What  is  the  purpose  of  gear  teeth  and  what  properties  must  they  pos- 
sess? 

4.  Define  cycloid,  hypocycloid  and  epicycloid. 

fa.  Draw  the  gear  teeth,  cycloidal  system,  for  a  gear  of  25  teeth,  2^  pitch 
with  3-in.  describing  circle,  (a)  for  a  spur  gear,  (6)  for  an  annular  gear. 

16.  A  pair  of  gears  are  to  connect  two  shafts  9  in.  apart,  ratio  4  to  5;  the 
diametral  pitch  is  to  be  2  and  the  describing  circles  1^  in.  diameter.  Draw 
the  teeth. 

7.  Define  the  various  terms  used  in  connection  with  gears  and  gear  teeth. 

8.  Define  involute  curve,  angle  of  obliquity,  base  circle.     Shoy  that  all 
involute  curves  from  the  same  base  circle  are  identical.  / 

V9.  Lay  out  the  gears  in  problem  6,  for  involute  teeth,  obliquity  14^°. 

10.  What  is  meant  by  interference  in  gear  teeth  and  what  is  the  cause  of 
it?     Why  does  it  occur  only  under  some  circumstances  and  not  always? 

11.  How  may  interference  be  prevented?     What  modification  is  usually 
made  in  rack  teeth? 

\12.  What  effect  has  variation  of  the  angle  of  obliquity  on  involute  teeth? 
13.  What  are  the  relative  merits  of  cycloidal  and  involute  teeth? 
\14.  Obtain   all  the   dimensions   of  the  following   gears:  (a)  Two   spur 
wheels,  velocity  ratio  2,  pitch  2,  shafts  9  in.  apart.     (6)  Outside  diameter 
of  gear  4  in.,  diametral  pitch  8.     (c)  Gear  of  50  teeth,  4  pitch,     (d)  Wheel 
of  103^  in.  outside  diameter,  40  teeth,     (e)  Pair  of  gears,  ratio  3  to  4, 
smaller  6  in.  pitch  diameter  with  30  teeth. 

16.  What  is  a  stub  tooth,  and  what  are  the  usual, -proportions?     What 
advantages  has  it? 

16.  Describe  the  various  methods  of  giving  the  sizes  of  gear  teeth  and 
find  the  relation  between  them. 

v  17.  Two  gears  for  an  automobile  are  to  hav^e.a  velocity  ratio  4  to  5,  shaft 
centers  4^  in.,  6  pitch;  draw  the  corree|  teeth jp  the  20°  stub  system. 
18.  Explain  the  construction  of  the  helical  ejfear  and  state  its  advantages. 


CHAPTER  VI 
BEVEL  AND  SPIRAL  GEARING 

104.  Gears  for  Shafts  not  Parallel. — Frequently  in  practice  the 
shafts  on  which  gears  are  placed  are  not  parallel,  in  which  case 
the  spur  gears  already  described  in  the  former  chapter  cannot 
be  used  and  some  other  form  is  required.     The  type  of  gearing 
used  depends,  in  the  first  place,  on  whether  the  axes  of  shafts 
intersect  or  not,  the  most  common  case  being  that  of  intersecting 
axes,  such  as  occurs  in  the  transmission  of  automobiles,  the  con- 
nection between  the  shaft  of  a  vertical  water  turbine  and  the 
main  horizontal  shaft,  in  governors,  and  in  very  many  other  well- 
known  cases. 

On  the  other  hand  it  not  infrequently  happens  that  the  shafts 
do  not  intersect,  as  is  true  of  the  crank-  and  camshafts  of  many 
gas  engines,  and  of  the  worm-gear  transmissions  in  some  motor 
trucks.  In  many  of  these  cases  the  shafts  are  at  right  angles,  as 
in  the  examples  quoted,  but  the  cases  where  they  are  not  are 
by  no  means  infrequent  and  the  treatment  of  the  present  chapter 
has  been  made  general. 

105.  Types  of  Gearing. — Where  the  axes  of  the  shafts  inter- 
sect the  gears  connecting  them  are  called  bevel  gears.     Where 
the  axes  of  the  shafts  do  not  intersect  the  class  of  gearing  depends 
upon  the  conditions  to  be  fulfilled  by  it.     If  the  work  to  be  done 
by  the  gearing  is  of  such  a  nature  that  point  contact  between  the 
teeth  is  sufficient,  then  screw  or  spiral  gearing  is  used,  a  form  of 
transmission  very  largely  used  where  the  shafts  are  at  rightangles, 
although  it  may  also  be  used  for  shafts  at  other  angles.      One 
peculiarity  of  this  class  is  that  the  diameters  of  the  gears  are  not 
determined  by  the  velocity  ratio  required,  and  in  fact  it  would 
be  quite  possible  to  keep  the  velocity  ratio  between  a  given  pair 
of  shafts  constant  and  yet  to  vary  within  wide  limits  the  relative 
diameters  of  the  two  gears  used. 

Where  it  is  desired  to  maintain  line  contact  between  the  teeth 
of  the  gears  on  the  two  shafts,  then  the  sizes  of  the  gears  are 
exactly  determined,  as  for  spur  gears,  by  the  velocity  ratio  and 

90 


BEVEL  AND  SPIRAL  GEARING 


91 


also  the  angle  and  distance  between  the  shafts.  Such  gears  are 
called  hyperboloidal  or  skew  bevel  gears  and  are  not  nearly  so 
common  as  the  spiral  gears,  but  are  quite  often  used. 

The  different  forms  of  this  gearing  will  now  be  discussed,  and 
although  a  general  method  of  dealing  with  the  question  might 
be  given  at  once,  it  would  seem  better  for  various  reasons  to 
defer  the  general  case  for  a  while  and  to  deal  in  a  special  way  with 
the  simpler  and  more  common  case  afterward  giving  the  general 
treatment. 

BEVEL  GEARING 

106.  Bevel  Gearing. — The  first  case  is  where  the  axes  of  the 
shafts  intersect,  involving  the  use  of  bevel  gearing.  The  inter- 
secting angle  may  have  any  value  from  nearly  zero  to  nearly 
180°,  and  it  is  usual  to  measure  this  angle  on  the  side  of  the 
point  of  intersection  on 
which  the  bevel  gears  are 
placed.  A  very  common 
angle  of  intersection  is  90° 
and  if  in  such  a  case  both 
shafts  turn  at  the  same  speed 
the  two  wheels  would  be 
identical  and  are  then  called 
mitre  gears.  The  type  of 
bevel  gearing  corresponding 
to  annular  spur  gearing  is 
very  unusual  on  account  of 
the  difficulty  of  construction, 
and  because  such  gears  are 
usually  easily  avoided,  how- 
ever they  are  occasionally 
used. 

Let  A  and  B,  Fig.  54,  rep- 
resent the  axes  of  two  shafts  intersecting  at  the  point  C  at 
angle  6,  the  speeds  of  the  shafts  being,  respectively,  ni  and  n2 
revolutions  per  minute;  it  is  required  to  find  the  sizes  of  the 
gears  necessary  to  drive  between  them.  Let  E  be  a  point  cf 
contact  of  the  pitch  lines  of  the  desired  gears  and  let  its  dis- 
tances from  A  and  B  be  TI  and  r2,  these  being  the  respective 
radii.  Join  EC. 

Now  from  Sec.  83  it  will  be  seen  that  ntti  =  r2n2  since  the 


92 


THE  THEORY  OF  MACHINES 


pitch  circles  must  have  the  same  velocity,  there  being  no  .slipping 

between  them,  and  hence  —  =  -  -  is  constant,  that  is  at  any  point 

?"2       n\ 

where  the  pitch  surfaces  of  the  gears  touch,  —  must  be  constant, 

a  condition  which  will  be  fulfilled  by  any  point  on  the  line  EC. 
In  the  case  of  bevel  gearing,  therefore,  the  pitch  cylinders  re- 
ferred to  in  Sec.  83  will  be  replaced  by  pitch  cones  with  apex  at 
Cj  which  cones  are  generated  by  the  revolution  of  the  line  CE 
about  the  axes  A  and  B.  Two  pairs  of  frustra  are  shaded  in 
on  Fig.  54  and  both  of  these  would  fulfil  the  desired  conditions 
for  the  pitch  surfaces  of  the  gear  wheels,  so  that  in  the  case  of 
bevel  gearing  the  diameters  of  the  gears  are  not  fixed  as  with 
spur  wheels,  but  the  ratio  between  these  diameters  alone  is  fixed 
by  the  velocity  ratio  desired.  The  angles  at  the  apexes  of  the 
two  pitch  cones  are  20i  and  202  as  indicated.  If  6  =  90°  and 
HI  =  n2,  then  61  =  62  =  45°  and  this  gives  the  case  of  the  mitre 
gears. 

In  going  over  the  discussion  it  will  be  observed  that  when 
6  and  ni/nz  are  known,  the  angles  0i  and  02  and  hence  the  pitch 

cones  are  fully  determined 
but  the  designer  has  still 
the  option  of  selecting  one 
of  the  radii,  for  example 
7*1,  at  his  pleasure^  after 
which  the  other  one,  r2,  is 
determined. 

107.  Proper  Shape  of 
Teeth  on  Bevel  Gears.— 
It  is  much  beyond  the  pur- 
pose of  the  present  treatise 
to  go  into  a  discussion  of 
the  exact  method  of  ob- 
taining the  form  of  teeth 

for  bevel  wheels, 'because  such  a  method  is  indeed  complicated, 
and  the  practical  approximation  produces  very  accurate  teeth 
and  will  be  described.  Let  Fig.  55  represent  one  of  the  wheels 
with  angle  26,  at  the  vertex  of  the  pitch  cone  and  radii  r\  and  r/ 
selected  in  accordance  with  the  previous  discussion,  the  power 
to  be  transmitted,  and  other  details  fixed  by  the  place  in  which 
the  gear  is  to  be  used.  The  diameter  2r*i  is  the  pitch  diameter 


FIG.  55. — Bevel  gears. 


BEVEL  AND  SPIRAL  GEARING 


93 


of  the  gear,  while  the  distance  CF  is  called  the  cone  distance 
and  FH  the  back  cone  distance,  61  is  the  pitch  angle  and  other 
terms  are  the  same  as  are  used  for  spur  gears  and  already 
explained.  The  lines  DG  and  FH  are  normal  to  CF  and  in- 
tersect the  shaft  at  G  and  H  respectively. 

The  practical  method  is  to  make  the  teeth  at  F  the  same  shape 
and  proportions  as  they  would  be  on  a  spur  gear  of  radius  FH, 
while  at  D  they  correspond  to  the  teeth  on  a  spur  gear  of  radius 
DG,  and  a  similar  method  is  used  for  any  intermediate  point. 
The  teeth  should  taper  from  F  to  D  and  a  straight  edge  passing 
through  C  would  touch  the  tooth  at  any  point  for  its  entire  length. 
Either  the  involute  or  cycloidal  system  may  be  used. 


FIG.  56. — Spiral  tooth  bevel  gears. 

108.  Spiral  Tooth  Bevel  Gears. — Within  the  past  few  years  the 
Gleason  Works,  and  possibly  others,  have  devised  a  method  for 
cutting  bevel  gears  with  a  form  of  " spiral7'  tooth  of  the  same 
general  nature  as  the  helical  teeth  used  with  spur  gears  and  de- 
scribed at  Sec.  103.  A  cut  of  a  pair  of  these  from  a  photograph 
kindly  supplied  by  the  Gleason  Works  is  given  at  Fig.  56,  and 
shows  the  general  appearance  of  the  gears.  Nothing  appears  to 
be  gained  in  the  way  of  reducing  friction,  but  they  run  very 
smoothly  and  noiselessly  and  the  greater  steadiness  of  motion 
makes  them  of  value  in  automobiles  and  other  similar  machines, 
in  which  they  are  mainly  used  at  present. 


94 


THE  THEORY  OF  MACHINES 


HYPERBOLOIDAL  OR  SKEW  BEVEL  GEARING 
THE  TEETH  OF  WHICH  HAVE  LINE  CONTACT 

109.  Following  the  bevel  gearing  the  next  class  logically  is 
the  hyperboloidal  gearing  and  the  treatment  of  this  includes  the 
general  case  of  all  gearing  having  line  contact  between  the  teeth. 
Let  AO  and  BP,  Fig.  57,  represent  the  axes  of  two  shafts  which  are 
to  be  geared  together,  the  line  OP  being  the  shortest  line  between 
the  axes  and  is  therefore  their  common  perpendicular.  Let  the 
axes  of  the  shafts  be  projected  in  the  ordinary  way  on  two  planes, 
one  normal  to  OP  and  the  other  passing  through  OP  and  one  of 
the  axes  AO,  the  projections  on  the  former  plane  being  AO,  OP 
and  PB  while  those  on  the  second  plane  are  A'O',  O'P'  and  P'B'. 


TS 


UV 


r> 


PQ- 

nl 


A' 


—  h 


B' 


O 
p' 


FIG.  57. 


On  the  latter  plane  the  shaft  axes  will  appear  as  parallel  straight 
lines  with  O'P'  as  their  common  perpendicular,  while  on  the 
former  plane  OP  appears  as  a  single  point  where  AO  and  BP 
intersect.  The  angle  A  OB  =  6  is  the  angle  between  the  shafts 
and  the  distance  O'P'  is  the  distance  between  them  and  when  0 
and  O'P'  =  h  are  known  the  exact  positions  of  the  shafts  are 
given.  The  speeds  of  the  shafts  n\  and  nz  must  also  be  known, 
as  well  as  the  sense  in  which  they  are  to  turn. 

In  stating  the  angle  between  the  shafts  it  is  always  intended 
to  mean  the  angle  in  which  the  line  of  contact  must  lie,  thus  in 
Fig.  57  the  sense  of  rotation  would  indicate  that  the  line  of  con- 
tact CQ  must  lie  somewhere  in  the  angle  AOB  and  not  in  AiOB 
so  that  the  angle  B  =  A  OB  is  used  instead  of  A\OB.  Should 
the  shaft  AO  turn  in  the  opposite  to  that  shown,  then  the  line 


BEVEL  AND  SPIRAL  GEARING  95 

of  contact  would  be  in  the  angle,  AiOB,  such  as  CiQ  (since  an- 
nular gears  are  not  used  for  this  type)  and  then  the  angle  A  iOB 
would  be  called  9. 

110.  Data  Assumed. — It  is  assumed  in  the  problem  that  the 
angle  6,  the  distance  h,  and  the  speeds  n\  and  n2  or  the  ratio 
tti/7i2,  are  all  given  and  it  is  required  to  design  a  pair  of  gears  for 
the  shafts,  such  that  the  contact  between  the  teeth  shall  be  along 
a  straight  line,  the  gears  complying  with  the  above  data. 

111.  Determination  of  Pitch  Surfaces. — Let  the  line  of  contact 
of  the  pitch  surfaces  be  CQ  and  let  it  be  assumed  that  this  line 
passes  through  and  is  normal  to  OP,  so  that  on  the  right-hand 
projections  A'O',   C'Qr  and  B'P'  are  all  parallel.     The  problem 
then  is  to  locate  CQ  and  the  pitch  surfaces  to  which  it  corre- 
sponds, the  first  part  of  the  problem  being  therefore  to  deter- 
mine hi,  hzj  61  and  d2,  Fig.  57,  and  this  will  now  be  done. 

Select  any  point  R  on  CQ,  Fig.  57,  R  being  thus  one  point  of 
contact  between  the  required  gears,  and  from  R  drop  perpendicu- 
lars RT  and  RV  on  OA  and  BP  respectively.  These  perpendicu- 
lars, which  are  radii  of  the  desired  gears,  have  the  resolved  parts 
ST  and  UV,  respectively,  parallel  to  OP,  and  the  resolved  parts 
RS  and  R  U  perpendicular  both  to  OP  and  to  the  respective 
shafts.  These  resolved  parts  are  clearly  shown 'in  the  figure, 
and  a  most  elementary  knowledge  of  descriptive  geometry  will 
enable  the  reader  to  understand  their  locations.  Further,  it  is 
clear  the  RT2  =  RS*  +  ST'2  and  RV2  =  RU*  +  U'V*. 

At  the  point  of  contact  R,  the  correct  velocity  ratio  must  be 
maintained  between  the  shafts,  and  as  R  is  a  point  of  contact  it  is 
a  point  common  to  both  gears  From  the  discussion  in  Sec.  84 
it  will  be  clear  that,  as  a  point  on  the  gear  located  on  OA  the  motion 
of  R  in  a  plane  normal  to  the  line  of  contact  CQ  must  be  identical 
with  the  motion  of  the  same  point  R  considered  as  a  point  on 
the  wheel  on  BP,  that  is,  in  the  plane  normal  to  the  line  of  contact 
CQ,  the  two  wheels  must  have  the  same  motion  at  the  point  of 
contact  R.  Sliding  along  CQ  is  not  objectionable,  however, 
except  from  the  point  of  view  of  the  wear  on  the  teeth  and  causes 
no  uneyenness  of  motion  any  more  than  the  axial  motion  of 
spur  gears  would  do,  it  being  evident  that  the  endlong  motion  of 
spur  gears  will  in  no  way  affect  the  velocity  ratio  or  the  steadiness 
of  the  motion.  In  designing  this  class  of  gearing,  therefore,  no 
effort  is  made  to  prevent  slipping  along  the  line  of  contact  CQ. 

Imagine  now  that  the  motion  of  R  in  each  wheel  in  the  plane 


96  THE  THEORY  OF  MACHINES 

normal  to  CQ  is  divided  into  two  parts,  namely,  those  normal  to 
each  plane  of  reference  in  the  drawing.  Taking  first  the  motion 
of  R  parallel  to  OP  (that  is,  normal  to  the  first  plane)  and  in  the 
plane  normal  to  CQ,  its  motion  as  a  point  in  the  wheel  on  OA 
in  the  required  direction  is  proportional  to  RS  X  n\  and  as  a 
point  in  the  wheel  on  BP  its  motion  in  the  same  direction  is 
proportional  to  RU  X  n2.  So  that  the  first  condition  to  be 
fulfilled  is  that 

RS  X  ni  =  RU  X  n2 

But  RS  =  OR  sin  61  and  RU  =  OR  sin  02 

Hence  OR  sin  0i  X  m  =  OR  sin  02  X  n2 

or  ni  sin  0i  =  n2  sin  62 

In  the  second  place,  consider  the  motion  of  R  in  the  plane 
normal  to  CQ  but  in  the  direction  normal  to  OP.  As  a  point  in 
the  wheel  on  OA  its  motion  in  the  required  direction  is  propor- 
tional to  S'T'  X  ni  X  cos  0i,  while  as  a  point  in  the  wheel  on  PB 
its  motion  is  proportional  to  U'V  X  n2  X  cos  02. 

The  second  condition  therefore  is 

S'T'  XniX  cos  0i  =  U'V'  Xn2X  cos  02 
htfii  cos  0i  =  /i2n2  cos  02. 

112.  Equations  for  Finding  the  Line  of  Contact. — Two  other 
conditions  may  be  written  as  self-evident,  and  assembling  the 
four  sets  at  one  place,  for  convenience,  gives 

0i  +  02  =  0  (1) 

h1  +  h2  =  h  (2) 

HI  sin  0i  =  nz  sin  02  (3) 

and  hini  cos  0i  =  h2n2  cos  02  (4) 

These  four  equations  are  clearly  independent,  and  since  0,  h,  n\ 
and  n2  or  ni/n2  are  given,  the  values  of  0i,  02,  hi  and  h2  are  known 
and  hence  the  location  of  the  line  of  contact  CQ. 

113.  Graphical  Solution  for  Line  of  Contact. — The  most  simple 
solution  is  graphical  and  the  method  is  indicated  in  Fig.  58  where 
OA  and  OB  represent  the  projections  of  the  axes  of  the  shafts 
on  a  plane  normal  to  their  common  perpendicular.     Lay  off 
OM  and  ON  along  the  directions  of  OB  and  OA  respectively  to 
represent  to  any  scale  the  speeds  nz  and  HI,  if  the  latter  are  given 
absolutely,  but  if  not,  make  the  ratio  OM/ON  =  n2/ni  choosing 


BEVEL  AND  SPIRAL  GEARING 


97 


one  of  the  lines,  say  OM,  of  any  convenient  length.  It  is  im- 
portant to  lay  off  OM  and  ON  in  the  proper  sense,  and  since  the 
shafts  turn  in  opposite  sense,  these  are  laid  off  in  opposite  sense 
from  0.  Join  MN  and  draw  OK  perpendicular  to  M N.  Then 
NK/KM  =  fa/hi,  and  to  find  their  numerical  values  take  any 
distance  NL  to  represent  h,  join  LM,  and  draw  KJ  parallel 
LM.  Then  fa  =  JL,  fa  =  JN,  ONM  =  61  and  OMN  =  02. 


FIG.  58. 

The  proof  of  the  construction  is  as  follows:  Since  77^7-  =  - 

UN        n\ 

and  since  OK  =  OM  sin  OMK  =  ON  sin  ONK,  it  follows  that 
nz  _  OM  _  sin  ONK 
ni  "  ON  '"  sin  OMK' 

Comparing  this  with  equation   (3),  Sec.   112,  it  is  dear  that 

ONK  =  0!  and  OMK  =  02  so  that  OC  is  parallel  to  MN.     Again, 

NK  =  ON  cos  0i  and  M K  =  OM  cos  02  from  which 

NK    _  ON      cos  0i  _  HI     cos  0!  _  fa 

MK  ~  OM  '  cos  02  ~~  nz     cos  02  ~~  fa 


98 


THE  THEORY  OF  MACHINES 


by  comparing  with  equation  (4).     The  construction  for  finding 

the  numerical  values  of  hi  and  h2  requires  no  explanation. 

114.  General  and   Special  Cases. — A  few  applications  will 

show  the  general  nature  of  the  solution  found. 

Case  1. — Shafts  inclined  at  any  angle  0  and  at  distance  h 

apart.     This  is  the  general  case  already  solved  and  0i,  02,  hi 

and  h2  are  found  as  indicated. 

Case  2. — Shafts  inclined  at  angle  0  =  90°  and  at  distance  h 

apart.     Care  must  be  taken  not  to  confuse  the  method  and  type 

of  gear  here  described  with  the  spiral 
gear  to  be  discussed  later.  Choose  the 
axes  as  shown  in  Fig.  59,  lay  off  to 
scale  ON  =  HI  and  OM  =  n2  and  join 
MN',  then  draw  OK  perpendicular  to 
MN  from  which  (Sec.  113)  NK:KM 
=  h2:hi.  In  this'  case  hiUi  cos  0i  = 
h2n2  cos  02  gives 

hini       cos  02       sin  0i  n2 

— =  — —  = —  =  tan  0i  =  — , 

cos  0i       cos  0i  n\ 


and  hence 


hi        /^2\ 
h2        \nj 


To  take  a  definite  case,  suppose  n\ 
=  2n2  then 


N 


=    I/ 


FIG.  59. 


and  if  the  distance  apart  of  the  shafts; 
h,  is  20  in.  then  hi  =  4  in.  and  hz  =  16  in.,  and  the  angle  0i  is 
given  by 


or 


tan    0i  =  --  =K  = 


0!  =  26°  34'    and    02  =  90  -  6l  =  63°  26', 


so  that  the  line  of  contact  is  located. 

Case  3. — Parallel  shafts  at  distance  h  apart.  This  gives  the 
ordinary  case  of  the  spur  gear.  Here  0=0  and  therefore 
0i  =  0  =  02,  hence,  sin  0i  =  0  =  sin  02  and  cos  0i  =  1  =  cos  02, 


BEVEL  AND  SPIRAL  GEARING  99 

so  that  there  are  only  two  conditions  to  satisfy:  hi  +  h2  =  h 
and  hini  =  /i2n2.     Solving  these  gives 

^1      7 


and  substituting  in  hi  +  hz  =  /i  gives 


n2 


and 


formulas  which  will  be  found  to  agree  exactly  with  those  of  Sec. 
83  for  spur  gears. 

Case  4. — Intersecting  shafts.     Here  h  =  0,  therefore  hi  =  0 
and  h2  =  0.     Referring  to  Fig.  60,  draw  OM  =  nz  and  ON  =  HI 


FIG.  60. 

then  M N  is  in  the  direction  of  the  line  of  contact  OC.  There  are 
only  two  equations  here  to  satisfy:  0i  +  62  =  6  and  n\  sin  6\  = 
nz  sin  62  and  these  are  satisfied  by  MN.  Then  draw  OC  parallel 
to  MN  (compare  this  with  the  case  of  the  bevel  gear  taken  up  at 
the  beginning  of  the  chapter). 

Case  5. — Intersecting  shafts  at  right  angles.  Here  6  =  90°. 
Further  let  n2  =  HI  then  02  =  45°,  thus  the  wheels  would  be 
equal  and  are  mitre  wheels. 


100 


THE  THEORY  OF  MACHINES 


115.  Pitch  Surfaces. — Returning  to  the  general  problem  in 
which  the  location  of  the  line  of  contact  CQ  is  found  by  the 
method  described  for  finding  hi,  h%,  &i  and  02.  Now,  just  as  in 
the  case  of  the  spur  and  bevel  gears,  a  short  part  of  the  line  of 
contact  is  selected  to  use  for  the  pitch  surfaces  of  the  gears,  ac- 
cording to  the  width  of  face  which  is  decided  upon,  the  width  of 
face  largely  depending  upon  the  power  to  be  transmitted,  and 
therefore  being  beyond  the  scope  of  the  present  discussion. 

It  is  known  from  geometry  that  if  the  line  CQ  were  secured 
to  AO,  while  the  latter  revolved  the  former  line  would  describe  a 
surface  known  as  an  hyperboloid  of  revolution  and  a  second  hy  per- 


boloid  would  be  described  by  securing  the  line  CQ  to  BP,  the 
curved  lines  in  the  drawing,  Fig.  61,  showing  sections  of  these 
hyperboloids  by  planes  passed  through  the  axes  AO  and  BP. 
As  the  process  of  developing  the  hyperboloid  is  somewhat  diffi- 
cult and  long,  the  reader  is  referred  to  books  on  descriptive 
geometry  or  to  other  works  for  the  method.  In  the  solution  of 
such  problems  as  the  present  one,  however,  it  is  quite  unneces- 
sary to  draw  the  exact  forms  of  the  curves,  and  at  any  time  the 
true  radius  to  the  curve  maybe  computed  as  explained  at  Sec.  111. 
Or  referring  to  Fig.  57  the  radius  of  the  wheel  on  OA  at  the  point 
T  on  its  axis  is  computed  from  the  relation  RT2  =  RS2  -f  ST2 
where  RT  is  the  radius  sought.  In  this  way  any  number  of 
radii  may  be  computed  and  the  true  form  of  the  wheels  drawn  in. 


BEVEL  AND  SPIRAL  GEARING 


101 


Should  the  distance  h  be  small,  then  sections  of  the  hyper- 
boloids  selected  as  shown  at  D  and  E  must  be  used,  the  distances 
of  these  from  the  common  normal  depending  upon  the  size  of  the 
teeth  desired,  the  power  to  be  transmitted,  the  velocity  ratio, 
etc.,  in  which  case  true  curved  surfaces  will  have  to  be  used,  more 
especially  if  the  gears  are  to  have  a  wide  face.  If  the  face  is  not 
wide,  it  may  be  possible  to  substitute  frustra  of  approximately 
similar  conical  surfaces. 

If  the  distance  h  is  great  enough,  and  other  conditions  permit 
of  it,  it  is  customary  to  use  the  gorges  of  the  hyperboloids  as 
shown  at  F  and  G  and  where  the  width  of  gear  face  is  not  great, 
cylindrical  surfaces  may  often  be  substituted  for  the  true  curved 
surfaces.  For  the  wheels  F  and  G  the  angles  0i  and  02  give  the 
inclination  of  the  teeth  and  the  angles  of 
the  teeth  for  D  and  E  may  be  computed 
from  61  and  02. 

116.  Example.---To  explain  more  fully, 
take    Case    (2),  Sec.   114,  for  which    6  =' 
90°,  ^  =  2n2,  and    h  =  20   in.,    then   the 
formulas  give  hi  =  4  in.  and  h2  =  16  in. 
Let  it  be  assumed  that  the  drive  is  such  as 
to  allow  the  use  of  the  gorge  wheels  corres- 
ponding to  F  and  G,  then  the  wheel  on  OA 
will    have    a    diameter    di  =  2hi  =  8    in. 
and  that  on  BP  will  be  dz  =  32  in.  diame- 
ter.    Further  the  angles  have  been  determined  to  be  0i  =  26°  34', 
62  =  63°  26'.     As  the  numbers  of  teeth  will  depend  on  the  power 
transmitted,  etc.,  it  will  here  be  assumed  that  gear  F  has  ti  =  20 
teeth.     Then  the  circular  pitch,  measured  on  the  end  of  the  gear, 

7T    X   8 

from  center  to  center  of  teeth  along  the  pitch  line  is  pi  =  -  ^Q 

=  1.256  in.,  which  distance  will  be  different  from  the  correspond- 
ing pitch  in  the  other  gear  G  which  has  40  teeth.  As  the  gears 
are  to  work  together  the  normal  distance  from  center  to  center  of 
teeth  on  the  pitch  surface  must  be  the  same  in  each  gear.  This 
distance  is  called  the  normal  pitch,  and  is  the  shortest  distance 
from  center  to  center  of  teeth  measured  around  the  pitch  sur- 
face; it  is,  in  fact,  the  distance  from  center  to  center  of  the 
teeth  around  the  pitch  line  measured  on  a  plane  normal  to  the 
line  of  contact  (CQ  in  Fig.  57)  and  agrees  with  what  has  already 
been  said,  that  the  motion  in  this  normal  plane  must  the  be 


JL. 


102  THE  THEORY  OF  MACHINES 

same  in  each  gear.  Calling  the  normal  pitch  p,  then  for  both 
gears  p  =  pi  cos  0i  =  p2  cos  02  =  1.256  cos  26°  34'  =  1.123  in. 
For  the  gear  G  the  number  of  teeth  t%  =  40  since  n\  =  2n2  and 
pi  =  2.513  in.  while  p  =  1.123  in.  A  sketch  of  the  gear  F  is 
given  at  Fig.  62. 

117.  Form  of  Teeth. — Much  discussion  has  arisen  over  the 
correct  form  of  the  teeth  on  such  gears,  and  indeed  it  is  almost 
impossible  to  make  a  tooth  which  will  be  theoretically  correct, 
but  here  again  one  is  to  be  guided  by  the  fact  the  correct  condi- 
tions must  be  fulfilled  in  the  plane  normal  to  the  line  of  contact. 
Hence  on  this  normal  plane  the  teeth  should  have  the  correct 
involute  or  cycloidal  profile. 

In  this  type  of  gearing  there  is  a  good  deal  of  slip  along  the 
line  of  contact  (CQ)  resulting  in  considerable  frictional  loss  and 
wear,  but  such  gearing,  if  well  made  will  run  very  smoothly  and 
quietly.  Although  it  is  difficult  to  construct  there  are  cases 
where  the  positions  of  the  shafts  make  its  use  imperative. 

SPIRAL  OR  SCREW  GEARING 

THE  TEETH  OF  WHICH  HAVE  POINT  CONTACT 

118.  Screw  Gearing. — In  speaking  of  gears  for  shafts  which 
were  not  parallel  and  did  not  interest  two  classes  were  mentioned : 
(a)  hyperboloidal  gears,  and  (6)  spiral  or  screw  gears  and  this 
latter  class  will  now  be  discussed,  the  former  having  just  been 
dealt  with.     In  screw  gearing  there  is  no  necessary  relation 
between  the  diameters  of  the  wheels  and  the  velocity  ratio 
HI/ n2  between  the  shafts;  thus  it  is  frequently  found  that  while 
the  camshaft  of  a  gas  engine  runs  at  half  the  speed  of  the  crank- 
shaft, the  two  screw  gears  producing  the  drive  are  of  the  same 
diameters,  while  if  skew  level  gears  were  used  the  ratio  of  diam- 
eters would  be  1  to  4  (Sec.  114(cT))  and  bevel  and  spur  gears  for 
the  same  work  would  have  a  ratio  1  to  2. 

119.  Worm  Gearing. — The  most  familiar  form  of  this  gearing 
is  the  well-known  worm  and  worm  wheel  which  is  sketched  in 
Fig.  63,  and  it  is  to  be  noticed  that  the  one  wheel  takes  the  form 
of  a  screw,  this  wheel  being  distinguished  by  the  name  of  the 
worm.     The  distance  which  any  point  on  the  pitch  circle  of  the 
worm  wheel  is  moved  by  one  revolution  of  the  worm  is  called 
the  axial  pitch  of  the  worm,  and  if  this  pitch  corresponds  to  the 
distance  from  thread  to  thread  along  the  worm  parallel  to  its 


BEVEL  AND  SPIRAL  GEARING 


103 


axis,  the  thread  is  single  pitch.  If  the  distance  from  one  thread 
to  the  next  is  one-half  of  the  axial  pitch  the  thread  is  double 
pitch,  and  if  this  ratio  is  one-third  the  pitch  is  triple,  etc.  The 
latter  two  cases  are  illustrated  at  (a)  and  (6),  Fig.  64. 


Fig.  63. — Worm  gearing. 


(a) 


(b) 


FIG.  64. — Double  and  triple  pitch  worms. 


120.  Ratio  of  Gearing. — Let  pi  be  the  axial  pitch  of  the  worm 
and  D  be  the  pitch  diameter  of  the  wheel  measured  on  a  plane 
through  the  axis  of  the  worm  and  normal  to  the  axis  of  the  wheel. 


104  THE  THEORY  OF  MACHINES 

Then  the  circumference  of  the  wheel  is  ivD,  and  since,  by  defini- 
tion of  the  pitch,  one  revolution  of  the  worm  will  move  the  gear 

forward  pi  in.,  hence  there  will  be  —  revolutions  of  the  worm  for 

one  revolution  of  the  wheel,  or  this  is  the  ratio  of  the  gears. 
Let  t  be  the  number  of  teeth  in  the  gear,  then  if  the  worm  is  single 

pitch  t  =  —  or  the  ratio  of  the  gears  is  simply  the  number  of 

teeth  in  the  wheel.  If  the  worm  is  double  pitch,  then  pi  the 
distance  from  center  to  center  to  teeth  measured  as  before  is 
given  by  pi  =  2p',  where  p'  is  the  axial  distance  from  the  center 

of  one  thread  to  the  center  of  the  next  one,  and  t  =  — -  and  as 

irD 

the  ratio  of   the    gears. is  — ,  in  the  double  pitch  worm  this  is 

equal  to  ~  >  and  for  triple  pitch  it  is  -^>  etc. 

^  o 

121.  Construction  of  the  Worm. — A  brief  study  of  the  matter 
will  show  that  as  the  velocity  ratio  of  the  gearing  is  fixed  by  the 
pitch  of  the  worm  and  the  diameter  of  the  wheel,  hence  no  matter 
how  large  the  worm  may  be  made  it  is  possible  still  to  retain  the 
same  pitch,  and  hence  the  same  velocity  ratio,  for  the  same  worm 
wheel.  The  only  change  produced  by  changing  the  diameter  of 
the  worm  is  that  the  angle  of  inclination  of  the  spiral  thread  is 
altered,  being  decreased  as  the  diameter  increases,  and  vice  versa. 
The  angle  made  by  the  teeth  across  the  face  of  the  wheel  must 
be  the  same  as  that  made  by  the  spiral  on  the  worm,  and  if  the 
pitch  of  the  worm  be  denoted  by  pi  and  the  mean  diameter  of 
the  thread  on  the  latter  by  d,  then  the  inclination  of  the  thread 

is  given  by  tan  6  =  -—,>  and  this  should  also  properly  be  the  in- 

TTtt 

clination  of  the  wheel  teeth.  From  the  very  nature  of  the  case 
there  will  be  a  great  deal  of  slipping  between  the  two  wheels,  for 
while  the  wheel  moves  forward  only  a  single  tooth  there  will  be 
slipping  of  amount  wd,  and  hence  considerable  frictional  loss,  so 
that  the  diameter  of  the  worm  is  usually  made  as  small  as  possi- 
ble consistent  with  reasonable  values  of  6.  The  worm  is  often 
immersed  in  oil,  but  still  the  frictional  loss  is  frequently  above  25 
per  cent. 

When  both  the  worm  and  wheel  are  made  parts  of  cylinders, 
Fig.  65,  then  there  will  only  be  a  very  small  wearing  surface  on 
the  wheel,  but  as  this  is  unsatisfactory  for  power  transmission, 


BEVEL  AND  SPIRAL  GEARING 


105 


the  worm  and  wheel  are  usually  made  as  shown  in  section  in  the 
left-hand  diagram  in  Fig.  65  where  the  construction  increases 
the  wearing  surface.  The  usual  method  of  construction  is  to 
turn  the  worm  up  in  the  lathe,  cutting  the  threads  as  accurately 
as  may  be  desired,  then  to  turn  the  wheel  to  the  proper  outside 
finished  dimensions.  The  cutting  of  the  teeth  in  the  wheel  rim 
may  then  be  done  in  various  ways  of 
which  only  one  will  be  described,  that 
by  the  use  of  a  hob. 

122.  Worm  and  Worm-wheel  Teeth. 
— A  hob  is  constructed  of  steel  and  is 
an  exact  copy  of  the  worm  with  which 
the  wheel  is  to  work,  and  grooves  are 
cut  longitudinally  across  the  threads  so 
as  to  make  it  after  the  fashion  of  a 
milling  cutter;  the  hob  is  then  hardened  and  ground  and  is 
ready  for  service.  The  teeth  on  the  wheel  may  now  be  roughly 
milled  out  by  a  cutter,  after  which  the  hob  and  gear  are  brought 
into  contact  and  run  at  proper  relative  speeds,  the  hob  milling 
out  the  teeth  and  gradually  being  forced  down  on  the  wheel  till 
it  occupies  the  same  relative  position  that  the  worm  will  even- 


FIG.  65. 


FIG.  66. — Proportions  of  worrrO 

tually  take.     In  this  way  the  best  form  of  worm  teeth  are  cut 
and  the  worm  and  wheel  will  work  well  together. 

The  shape  of  teeth  on  the  worm  wheel  is  determined  by  the 
worm,  as  above  explained.  If  a  section  of  the  worm  is  taken 
by  a  plane  passing  through  its  axis,  the  section  of  the  threads  is 
made  the  same  as  that  of  the  rack  for  an  ordinary  gear,  and  more 
usually  the  involute  system  is  used  with  an  angle  14J^°.  A  sec- 


106  THE  THEORY  OF  MACHINES 

tion  of  the  worm  thread  is  shown  at  Fig.  66  in  which  the  propor- 
tions used  by  Brown  and  Sharpe  are  the  same  as  in  a  rack. 

123.  Large  Ratio  in  this  Gearing. — Although  the  frictional 
losses  in  screw  gearing  are  large,  even  when  the  worm  works 
immersed  in  oil,  yet  there  are  great  advantages  in  being  able  to 
obtain   high   velocity  ratios   without   excessively  large   wheels. 
Thus  if  a  worm  wheel  has  40  teeth,  and  is  geared  with  a  single- 
threaded  worm,  the  velocity  ratio  will  be  jr •  while  with  a  double- 

2         1 
threaded  worm  it  will  be  jr  =  ^  so  that  it  is  very  convenient 

for  large  ratios.  It  also  finds  favor  because  ordinarily  it  cannot 
be  reversed,  that  is,  the  worm  must  always  be  used  as  the  driver 
and  cannot  be  driven  by  the  wheel  unless  the  angle  8  is  large. 
In  cream  separators,  the  wheel  is  made  to  drive  the  worm. 

124.  Screw  Gearing. — Consider  now  the  case  of  the  worm  and 
wheel  shown  in  Fig.  65,  in  which  both  are  cylinders,  and  suppose 
that  with  a  worm  of  given  size  a  change  is  made  from  a  single 
to  double  thread,  at  the  same  time  keeping  the  threads  of  the 
same  size.     The  result  will  be  that  there  will  be  an  increase  in 
the  angle  6  and  hence  the  threads  will  run  around  the  worm  and 
the  teeth  will  run  across  the  wheel  at  greater  angle  than  before. 
If  the  pitch  be  further  increased  there  is  a  further  increase  in  6 
and  this  may  be  made  as  great  as  45°,  or  even  greater,  and  if  at 
the  same  time  the  axial  length  of  the  worm  be  somewhat  de- 
creased, the  threads  will  not  run  around  the  worm  completely, 
but  will  run  off  the  ends  just  in  the  same  way  as  the  teeth  of 
wheels  do. 

By  the  method  just  described  the  diameter  of  the  worm  is  un- 
altered, and  yet  the  velocity  ratio  is  gradually  approaching  unity, 
since  the  pitch  is  increasing,  so  that  keeping  to  a  given  diameter 
of  worm  and  wheel,  the  velocity  ratio  may  be  varied  in  any  way 
whatever,  or  the  velocity  ratio  is  independent  of  the  diameters 
of  the  worm  and  wheel.  When  the  pitch  of  the  worm  is  increased 
and  its  length  made  quite  short  it  changes  its  appearance  from 
what  it  originally  had  and  takes  the  form  of  a  gear  wheel  with 
teeth  running  in  helices  across  the  face.  A  photograph  of  a  pair 
of  these  wheels  used  for  driving  the  camshaft  of  a  gas  engine  is 
shown  in  Fig.  67,  and  in  this  case  the  wheels  give  a  velocity  ratio 
of  2  to  1  between  two  shafts  which  do  not  intersect,  but  have  an 
angle  of  90°  between  planes  passing  through  their  axes.  This 


BEVEL  AND  SPIRAL  GEARING 


107 


form  of  gear  is  very  extensively  used  for  such  purposes  as  afore- 
said, giving  quiet  steady  running,  but,  of  course,  the  frictional 
loss  is  quite  high. 

Some  of  the  points  mentioned  may  be  made  clearer  by  an  illus- 
tration. Let  it  be  required  to  design  a  pair  of  gears  of  this 
type  to  drive  the  camshaft  of  a  gas  engine  from  the  crankshaft, 
the  velocity  ratio  in  this  case  being  1  to  2,  and  let  both  gears  be 
of  the  same  diameter,  the  distance  between  centers  being  12  in. 
From  the  data  given  the  pitch  diameter  of  each  wheel  will  be 


FIG.  67. — Screw  gears. 

12  in.  and  since  for  one  revolution  of  the  camshaft  the  crankshaft 
must  turn  twice,  the  pitch  of  the  thread  on  the  worm  must  be 
MX  TrX  12  =  18.85  in.  For  the  gear  on  the  crankshaft  (cor- 
responding to  the  worm)  the  " teeth"  will  run  across  its  face  at 

18  85 
an  angle  given  by  tan  6  =  -    '  10  =  0.5,  or  0  =  26°  34',  and  this 

7T  X   1^ 

angle  is  to  be  measured  between  the  thread  or  tooth  and  the  plane 
normal  to  the  axis  of  rotation  of  the  worm  (see  Fig.  64).  The 
angle  of  the  teeth  of  the  gear  on  the  camshaft  (corresponding  to 


108  THE  THEORY  OF  MACHINES 

the  worm  wheel)  will  be  90  -  26°  34'  =  63°  26'  measured  in  the 
same  way  as  before  (compare  this  with  the  gear  in  Sec.  116). 

It  will  be  found  that  the  number  of  teeth  in  one  gear  is  double 
that  in  the  other,  also  the  normal  pitch  of  both  gears  must  be 
the  same.  The  distance  between  adjacent  teeth  is  made  to  suit 
the  conditions  of  loading  and  will  not  be  discussed. 

Spiral  gearing  may  be  used  for  shafts  at  any  angle  to  one 
another,  although  they  are  most  common  in  practice  where  the 
angle  is  90°.  A  more  detailed  discussion  of  the  matter  will  not 
be  attempted  here  and  the  reader  is  referred  to  other  complete 
works  on  the  subject. 

125.  General  Remarks  on  Gearing. — In  concluding  this  chap- 
ter it  is  well  to  point  out  the  differences  in  the  two  types  of  gear- 
ing here  discussed.  In  appearance  in  many  cases  it  is  rather  diffi- 
cult to  tell  the  gears  apart,  but  a  close  examination  will  show 
the  decided  difference  that  in  hyperboloidal  gearing  contact  be- 
tween the  gears  is  along  a  straight  line,  while  in  spiral  gearing 
contact  is  at  a  point  only.  A  study  of  gears  which  have  been  in 
operation  shows  this  clearly,  the  ordinary  spiral  gear  as  used  in 
a  gas  engine  wearing  only  over  a  very  small  surface  at  the  centers 
of  the  teeth.  It  is  also  to  be  noted  that  the  teeth  of  hyper- 
boloidal gears  are  straight  and  run  perfectly  straight  across  the 
face  of  the  gear,  while  the  teeth  of  spiral  gears  run  across  the  face 
in  helices. 

Again  in  both  classes  of  gears  where  the  spiral  gears  have  the 
form  shown  at  Fig.  67,  the  ratio  between  the  numbers  of  teeth  on 
the  gear  and  pinion  is  the  velocity  ratio  transmitted,  but  in  the 
case  of  the  spiral  gears  the  relative  diameters  may  be  selected 
as  desired,  while  in  the  hyperboloidal  gears  the  diameters  are 
fixed  when  the  angle  between  the  shafts  and  the  velocity  ratio 
is  given. 

QUESTIONS  ON  CHAPTER  VI 

1.  Two  shafts  intersect  at  80°,  one  running  at  double  the  speed  of  the 
other.     Design  bevel  gears  for  the  purpose,  the  minimum  number  of  teeth 
being  12  and  the  diametral  pitch  3. 

2.  What  would  be  the  sizes  of  a  pair  of  miter  gears  of  20  teeth  and  1>^  iia. 
pitch?     If  the  face  is  two  and  one-half  times  the  pitch,  find  the  radii  of  the 
spur  gears  at  the  two  ends,  from  which  the  teeth  are  determined. 

3.  Two  shafts  cross  at  angle  6  =  45°  and  are  10  in.  apart,  velocity  ratio  2, 
locate  the  line  of  contact  of  the  teeth. 

4.  Find  the  diameters  of  a  pair  of  gorge  wheels  to  suit  question  3;  also 
the  sizes  of  the  gears  if  the  distance  OS  =  12  in. 


BEVEL  AND  SPIRAL  GEARING  109 

5.  If  the  angle  between  the  shafts  is  90°  in  the  above  case,  find  the  sizes 
of  the  gorge  wheels. 

6.  Explain  fully  the  difference  between  spiral  and  skew  bevel  gears. 

7.  A  worm  gear  is  to  be  used  for  velocity  ratio  of  100,  the  worm  to  be  6  in. 
diameter,  1%  in.  pitch,  and  single-thread;  find  the  size  of  the  gear  and  the 
angle  of  the  teeth. 

8.  What  would  be  the  dimensions  above  for  a  double-threaded  worm  ? 


CHAPTER  VII 
TRAINS  OF  GEARING 

126.  Trains  of  Gearing. — In  ordinary  practice  gears  are  usually 
arranged  in  a  series  on  several  separate  axles,  such  a  series  being 
called  a  train  of  gearing,  so  that  a  train  of  gearing  consists  of 
two  or  more  toothed  wheels  which  all  have  relative  motion  at  the 
same  time,  the  relative  angular  velocities  of  all  wheels  being 
known  when  that  of  any  one  is  given.     A  train  of  gearing  may 
always  be  replaced  by  a  single  pair  of  wheels  of  suitable  diame- 
ters, but  frequently  the  sizes  of  the  two  gears  are  such  as  to 
make  the  arrangement  undesirable  or  impracticable. 

When  the  train  consists  of  four  or  more  wheels,  and  when  two 
of  these  of  different  sizes  are  keyed  to  the  same  intermediate 
shaft,  the  arrangement  is  a  compound  train.  This  agrees  with 
the  definition  of  a  compound  chain  given  in  Chapter  I,  because 
one  of  the  links  contains  over  two  elements,  this  being  the  pair 
of  gears  on  the  intermediate  shaft.  The  compound  trains  are 
in  very  common  use  and  are  sometimes  arranged  so  that  the  axes 
of  the  first  and  last  gears  coincide,  in  which  case  the  train  is  said 
to  be  reverted ;  a  very  common  illustration  of  this  is  the  train  of 
gears  between  the  minute  and  hour  hands  of  a  clock,  the  axes 
of  both  hands  coinciding. 

127.  Kinds  of  Gearing  Trains. — If  one  of  the  gears  in  the  train 
is  prevented  from  turning,  or  is  held  stationary,  and  all  of  the  other 
gears  revolve  relatively  to  it,  usually  by  being  carried  bodily 
about  the  fixed  gear  as  in  the  Weston  triplex  pulley  block  or 
the  differential  on  an  automobile  when  one  wheel  stops  and  the 
other  spins  in  the  mud,  the  arrangement  is  called  an  epicyclic 
train.     Such  a  train  may  be  used  as  a  simple  train  of  only  two 
wheels,  but  is  much  more  commonly  compounded  and  reverted 
so  that  the  axis  of  the  last  wheel  coincides  with  that  of  the  first. 

For  the  ordinary  train  of  gearing  the  velocity  ratio  is  the  num- 
ber of  turns  of  the  last  wheel  divided  by  the  number  of  turns 
of  the  first  wheel  in  the  same  time,  whereas  in  the  epicyclic  train 
the  velocity  ratio  is  the  number  of  turns  of  the  last  wheel  in  the 

110 


TRAINS  OF  GEARING  111 

train  divided  by  the  number  of  turns  in  the  same  time  of  the 
frame  carrying  the  moving  wheels. 

128.  Ordinary  Trains1  of  Gearing. — It  will  be  well  to  begin 
this  discussion  with  the  most  common  class  of  gearing  trains, 
that  is  those  in  which  all  the  gears  in  the  train  revolve,  and  the 
frame  carrying  their  axles  remains  fixed  in  space.  The  outline 
of  such  a  train  is  shown  in  Fig.  68,  where  the  frame  carrying  the 
axles  is  shown  by  a  straight  line  while  only  the  pitch  circles  of 
the  gears  are  drawn  in;  there  are  no  annular  gears  in  the  train 
shown,  although  these  may  be  treated  similarly  to  spur  gears. 
Let  HZ  be  the  number  of  revolutions  per  minute  made  by  the 


FIG.  68. 

last  gear  and  n\  the  corresponding  number  of  revolutions  per 
minute  made  by  the  first  gear,  then,  from  the  definition  already 
given,  the  ratio  of  the  train  is 

n2       the  number  of  revolutions  per  minute  of  the  last  gear 
n\       the  number  of  revolutions  per  minute  of  the  first  gear. 

The  figure  shows  a  train  consisting  of  six  spur  gears  marked  1, 
a,  b,  c,  e  and  2,  and  let  1  be  considered  the  first  gear  and  2  the 
last  gear. 

The  following  notation  will  be  employed :  HI,  na  =  nb,  nc  =  ne 
and  n2  will  represent  the  revolutions  per  minute,  n,  ra,  n,  rc,  re 
and  r2  the  radii  in  inches,  and  t\}  ta,  tb,  tc,  te  and  t%  the  numbers  of 
teeth  for  the  several  gears  used.  The  gears  a  and  b  and  also  the 
gears  c  and  e  are  assumed  to  be  fastened  together,  so  that  the 
train  is  compounded,  a  statement  true  of  any 'train  of  over  two 
gears.  Any  pair  of  gears  such  as  6  and  c,  which  mesh  with  one 
another,  must  have  the  same  type  and  pitch  of  teeth,  but  both 
the  type  and  pitch  may  be  different  for  any  other  pair  which  mesh 
together,  such  as  e  and  2;  the  only  requirement  is  that  each  gear 

1In  what  follows  in  this  chapter  reference  is  made  to  spur  and  bevel 
gears  only. 


112  THE  THEORY  OF  MACHINES 

must  have  teeth   corresponding  with   those  in  the  gear  with 
which  it  meshes. 

129.  Ratio  of  the  Train. — Now,  from  the  results  given  in  Sec. 
101,  evidently 

na       TI       ti  nc       Tb       U 

—  =  —  =  — ,  ana  —  =  —  =  - 

n\          Ta          la  Ub          Tc  tc 

also 

—  =  —  =  -. 

ne  ~  TZ  ~  tz 

Therefore,  the  ratio  of  the  train 

n\       n\    na    nc 

na    nc    nz    .  , 

=  —  —  —  since  na  =  nb  and  nc  =  n. 
HI    nb    ne 

and  therefore 

R  =  r-l  X  T-  X  -  =  r-~^~- 

T a          Tc          TZ          Ta   X  Tc  X  TZ 
__ti          tb          te          tiXtbXte 

i        /N      ,        /N      /       


ta  tc  t2  taXtcX  tz 

Calling  the  first  wheel  in  each  pair  (i.e.,  1,  b  and  e)  the  driver, 
then  the  formula  for  the  ratio  of  the  train  may  be  written  thus, 
The  ratio  of  any  gear  train  is  the  product  of  the  radii  of  the  drivers 
divided  by  the  product  of  the  radii  of  the  driven  wheels,  or  the 
ratio  of  the  train  is  the  product  of  the  numbers  of  teeth  in  the 
drivers  divided  by  the  product  of  the  numbers  of  teeth  in  the 
driven  wheels. 

The  same  law  may  be  readily  shown  to  apply  although  some  of 
the  gears  are  annular,  and  indeed  is  true  when  a  pair  of  gears  is 
replaced  by  an  open  or  a  crossed  belt  or  a  pair  of  sprockets  and 
a  chain. 

To  take  an  illustration,  let  the  train  shown  in  Fig.  68  have 
gears  of  the  following  sizes: 

7*1  =  6  in.,  ra'=  3%  in.,  n  =  4  in.  rc  =  2%  in.,  re  =  5  in.  and 
7*2  =  3  in.  and  let  the  diametral  pitches  be  4,  6  and  8  for  the  pairs 
1  and  a,  b  and  c,  and  e  and  2  respectively.  Then  ti  =  48,  ta  =  30, 
tb  =  48,  tc  =  32,  te  =  "80  and  t2  =  48  teeth. 

The  ratio  of  the  train  is  then 

6X4X5 
R  =  3%  X  2%  X  3  =  4  fr°m  the 


TRAINS  OF  GEARING  113 

or 

48  X  48  X  80 
=  QQ  y  QO  X  48  =      fr°m  the  numbers  of  teeth. 

Further,  if  wheel  1  turn  at  a  speed  of  HI  =  50  revolutions  per 
minute  the  speeds  of  the  other  gears  will  be  na  =  nb  =  80  revolu- 
tions, nc  =  ne  =  120  revolutions  and  n2  =  200  revolutions  per 
minute. 

If  the  distance  between  the  axes  of  gears  1  and  2  were  fixed 
by  some  external  conditions  at  the  distance  apart  corresponding 
to  the  above  train,  then  the  whole  train  could  be  replaced  by  a 
pair  of  gears  having  radii  of  19.53  in.  and  4.88  in.,  and  these  would 
give  the  same  velocity  ratio  as  the  train,  but  would  often  be 
objectionable  on  account  of  the  large  size  of  the  larger  gear. 

The  sense  of  rotation  of  the  various  gears  may  now  be  ex- 
amined. Looking  again  at  Fig.  68,  it  is  observed  that  for  two 
spur  wheels  (which  have  one  contact)  the  sense  is  reversed,  where 
there  are  two  contacts,  as  between  1  and  c  the  sense  remains 
unchanged,  and  with  three  contacts  such  as  between  1  and  2 
the  sense  is  reversed  and  the  rule  for  determining  the  relative 
sense  of  rotation  of  the  first  and  last  wheels  may  be  stated  thus: 
In  any  spur-wheel  train,  if  the  number  of  contacts  between  the 
first  and  last  gears  are  even  then  both  turn  in  the  same  sense, 
and  if  the  number  of  contacts  is  odd,  then  the  first  and  last 
wheels  turn  in  the  opposite  sense.  Should  the  train  contain 
annular  gears,  the  same  rule  will  apply  if  it  is  remembered  that 
any  contact  with  an  annular  gear  has  the  same  effect  as  two 
contacts  between  spur  gear.  The  same  rule  also  applies  in  case 
belts  are  used,  an  open  belt  corresponding  to  an  annular  gear  and 
a  crossed  one  to  spur  gears. 

The  rules  both  for  ratio  and  sense  of  rotation  are  the  same  for 
bevel  gears  as  for  spur  gears. 

130.  Idlers. — It  not  infrequently  happens  that  in  a  compound 
train  the  two  gears  on  an  intermediate  axle  are  made  of  the  same 
size  and  combined  into  one;  thus  ra  may  be  made  equal  to  7-5  or 
ta  =  tb.  This  single  intermediate  wheel,  then,  has  no  effect  on 
the  velocity  ratio  R,  as  an  inspection  of  the  formula  for  R  will 
show,  and  is  therefore  called  an  idler.  The  sole  purpose  in  using 
such  wheels  is  either  to  change  the  sense  of  rotation  or  else  to 
increase  the  distance  between  the  centers  of  other  wheels  without 
increasing  their  diameters. 


114  THE  THEORY  OF  MACHINES 

131.  Examples.  —  The  application  of  the  formula  may  be  best 
explained  by  some  examples  which  will  now  be  given: 

1.  A  wheel  of  144  teeth  drives  one  of  12  teeth  on  a  shaft  which 
makes  one  revolution  in  12  sec.,  while  a  second  one  driven  by  it 
turns  once  in  5  sec.     On  the  latter  shaft  is  a  40-in.  pulley  con- 
nected by  a  crossed  belt  to  a  12-in.  pulley;  this  latter  pulley  turns 
twice  while  one  geared  with  it  turns  three  times.     Show  that  the 
ratio  is  144  and  that  the  first  and  last  wheels  turn  in  the  same 
sense. 

2.  It  is  required  to  arrange  a  train  of  gearing  having  a  ratio 
,250 

ofir 

It  is  possible  to  solve  this  problem  by  using  two  gears  having 
250  teeth  and  13  teeth  respectively,  but  in  general  the  larger 
wheel  will  be  too  big  and  it  will  be  well  to  make  up  a  train  of  four 

250 
or  six  gears.     Break  the  ratio  up  into  factors,  thus:  R  =  -y^-  = 

50       60 

75-  X  TT;  and  referring  to  the  formula  for  the  ratio  of  a  train  it 

lO  \.a 

is  evident  that  one  could  be  made  up  of  four  gears  having  50 
teeth,  13  teeth,  60  teeth  and  12  teeth,  and  these  would  be  ar- 
ranged with  the  first  wheel  on  the  first  axle,  the  gears  of  60  teeth 
and  13  teeth  would  be  keyed  together  and  turn  on  the  inter- 
mediate axle  and  the  12-tooth  gear  would  be  on  the  last  axle  and 
the  contacts  would  be  the  50  to  the  13  and  the  60  to  the  12-tooth 
wheel. 

Evidently,  the  data  given  allow  of  a  great  many  solutions  for 
this  problem,  another  with  six  wheels  being, 


=  5540601540 

13    ""  1  A  4  A  13  "  12  A  12  A  13 

This  would  give  a  train  similar  to  Fig.  68  in  which  the  gears  are 
ti  =  60,  ta  =  12,  tb  =  15,  te  =  12,  te  =  40  and  t2  =  13  teeth. 

3.  To  design  a  train  of  wheels  suitable  for  connecting  the  sec- 
ond hand  of  a  watch  to  the  hour  hand.  Here  the  ratio  is 
R  =  720  and  the  first  and  last  wheels  must  turn  in  the  same 
sense,  and  as  annular  wheels  are  not  used  for  this  purpose,  the 
number  of  contacts  must  be  even.  The  following  two  solutions 
would  be  satisfactory  for  eight  wheels  : 

R  -  700  -  4X4X5X9  -  §?  v  48       50       108 
"      °  '  1  '  14  X  12  X  10  X   12 


TRAINS  OF  GEARING  115 

p  -  yon  -  6X6X4X5  -Z?v?0fi280 

1  "  12  A  10  X  13  X  12 

Attention  is  called  to  the  statement  in  Sec.  89  that  it  is  un- 
usual to  have  wheels  of  less  than  12  teeth. 

4.  Required  the  train  of  gears  suitable  for  connecting  the  min- 
ute and  hour  hands  of  a  clock. 

Here  R  =  12  and  the  train  must  be  reverted;  further,  since 
both  hands  turn  in  the  same  sense  there  must  be  an  even  number 
of  contacts,  and  four  wheels  will  be  selected.  In  addition  to 
obtaining  the  correct  velocity  ratio  it  is  necessary  that  TI  +  ra  — 
tb  +  7*2,  and  if  all  the  wheels  have  the  same  pitch  ti  -f  ta  =  h  +  fa- 
The  following  train  will  evidently  produce  the  correct  result: 

„       19      4X3       48       46 
~~T    =  12XW 

or  ti  =  48,  ta  =  12,  tb  =  45  and  t2  =  15  teeth,  the  hour  hand 
carrying  the  48  teeth  and  the  minute  hand  the  15  teeth. 

132.  Automobile  Gear  Box. — Very  many  applications  of 
trains  of  gearing  have  been  made  to  automobiles  and  a  drawing 
of  a  variable-speed  transmission  is  shown  in  Fig.  69.  The  draw- 
ing shows  an  arrangement  for  three  forward  speeds  (one  without 
using  the  gears)  and  one  reverse.  The  shaft  E  is  the  crankshaft 
and  to  it  is  secured  a  gear  A  having  also  a  part  of  a  jaw  clutch  B 
on  its  right-hand  side.  Gear  A  meshes  with  another  G  on  a 
countershaft  M,  which  carries  also  the  gears  H,  J  and  K  keyed 
to  it,  and  whenever  the  engine  shaft  E  operates  the  gears  G,  H, 
J  and  K  are  running.  On  the  right  is  shown  the  power  shaft  P 
which  extends  back  to  the  rear  or  driving  axle;  this  shaft  is 
central  with  E  and  carries  the  gears  D  and  F  and  also  the  inner 
part  C  of  the  jaw  clutch  for  B. 

The  gears  D  and  F  are  forced  to  rotate  along  with  P  by  means 
of  keys  at  T,  but  both  gears  may  be  slid  along  the  shaft  by  means 
of  the  collars  at  N  and  R  respectively.  In  addition  to  the  gears 
already  mentioned  there  is  another,  L,  which  always  meshes  with 
K  and  runs  on  a  bearing  behind  the  gear  K. 

Assuming  the  driver  wishes  to  operate  the  car  at  maximum 
speed  he  throws  F  into  the  position  shown  and  pushes  D  to  the 
left  so  that  the  clutch  piece  C  engages  with  B,  in  which  case  P 
runs  at  the  same  speed  as  the  -engine  shaft  E.  Second  highest 
speed  is  obtained  by  slipping  D  to  the  right  until  it  comes  into 


116  THE  THEORY  OF  MACHINES 

contact  with  H,  the  ratio  of  gears  then  being  A  to  G  and  H  to  D; 
F  remains  as  shown.  For  lowest  speed  D  is  placed  as  shown  in 
the  figure  and  F  slid  into  contact  with  J;  the  shaft  P  and  the  car 
are  reversed  by  moving  F  to  the  right  until  it  meshes  with  L, 
the  gear  ratio  being  A  to  G  and  K  to  L  to  F. 

Builders  of  automobiles  so  design  the  operating  levers  that  it  is 
possible  to  have  only  one  set  of  gears  in  operation  at  once. 


FIG.  69. — Automobile  gear  box. 

133.  The  Screw-cutting  Lathe. — Most  lathes  are  arranged  for 
cutting  threads  on  a  piece  of  work,  and  as  this  forms  a  very 
interesting  application  of  the  principles  already  described,  it  will 
be  used  as  an  illustration. 

The  general  arrangement  of  the  headstock  of  a  lathe  is  shown 
in  Fig.  70,  and  in  this  case  in  order  to  make  the  present  discussion 
as  simple  as  possible,  it  is  assumed  that  the  back  gear  is  not  in 
use.  The  cone  C  is  connected  by  belt  to  the  countershaft  which 
supplies  the  power,  the  four  pulleys  permitting  the  operation  of 
the  lathe  at  four  different  speeds.  This  cone  is  secured  to  the 
spindle  S,  which  carries  the  chuck  K  to  which  the  work  is  at- 


TRAINS  OF  GEARING 


117 


tached  and  by  which  it  is  driven  at  the  same  rate  as  the  cone  C. 
On  the  other  end  of  S  is  a  gear  e,  which  drives  the  gear  h  through 
one  idler  g  or  two  idlers  /  and  g.  The  shaft  which  carries  h 
also  has  a  gear  1  which  is  keyed  to  it,  and  must  turn  with  the  shaft 
at  the  same  speed  as  h.  The  gear  1  meshes  with  a  pinion  a  on  a 
separate  shaft,  this  pinion  being  also  rigidly  connected  to  and  re- 
volving with  gear  b,  which  latter  gear  meshes  with  a  wheel  2 
keyed  to  the  leading  screw  L.  Thus  the  spindle  S  is  geared  to 
the  leading  screw  L  through  the  wheels  e,  /,  g,  h,  1,  a,  6,  2  of 
which  the  first  four  are  permanent,  while  the  latter  four  may 
be  changed  to  suit  conditions,  and  are  called  change  gears. 

The  work  is  attached  to  the  chuck  K  on  S  and  is  supported  by 
the  center  on  the  tail  stock  so  that  it  rotates  with  K.     The  lead- 


Chuck 


FIG.  70. — Lathe  head  stock. 

ing  screw  L  passes  through  a  nut  in  the  carriage  carrying  the 
cutting  tool,  and  it  will  be  evident  that  for  given  gears  on  1, 
a,  b,  2  a  definite  number  of  turns  of  S  correspond  to  a  definite 
number  of  turns  of  L,  and  hence  to  a  certain  horizontal  travel 
of  the  carriage  and  cutting  tool.  Suppose  that  it  is  desired  to 
cut  a  screw  on  the  work  having  s  threads  per  inch,  the  number  of 
threads  per  inch  I  on  the  leading  screw  being  given.  This  requires 
that  while  the  tool  travels  1  in.  horizontally,  corresponding  to  I 
turns  of  the  leading  screw  L,  the  work  must  revoive  s  times,  or 
if  HI  represents  the  revolutions  per  minute  of  the  work,  and  n2 
those  of  the  leading  screw,  then 

_n2  _l  _te       ti       k 

""* *  *"~  ~~~     j        /N     •       ^\     j 

n\      s      th      ta      tz 


118  THE  THEORY  OF  MACHINES 

where  te,  fo,  etc.,  represent  the  numbers  of  teeth  in  the  gears. 
Evidently  /  and  g  are  idlers  and  have  no  effect  on  the  ratio. 
In  many  lathes  the  gears  e  and  h  are  made  the  same  size  so 
that  gear  1  turns  at  the  same  speed  as  gear  h. 

Then  R  =  ^xr 

ta  tz 

This  ratio  is  used  in  the  example  here.  For  many  purposes 
also  ta  =  tb. 

Further,  if  L  and  S  turn  in  the  same  sense,  and  if  the  leading 
screw  has  a  right-hand  thread,  as  is  usual,  then  the  thread  cut 
on  the  work  will  also  be  right-hand.  The  idlers  /  and  g  arc 
provided  to  facilitate  this  matter,  and  if  a  right-hand  thread 
is  to  be  cut,  the  handle  m  carrying  the  axes  of  /  and  g  is  moved 
so  that  g  alone  connects  e  and  h,  while,  if  a  left-hand  thread  is 
to  be  cut  the  handle  is  depressed  so  that  /  meshes  with  e  and  g 
with  h.  The  figure  shows  the  setting  for  a  right-hand  thread. 

An  illustration  will  show  the  method  of  setting  the  gears  to  do 
a  given  piece  of  work.  Suppose  that  a  lathe  has  a  leading  screw 
cut  with  4  threads  per  inch,  and  the  change  gears  have  respect- 
ively 20,  40,  45,  50,  55,  60,  65,  70,  75,  80  and  115  teeth.  Assume 

te    ==    th- 

1.  It  is  required  to  cut  a  right-hand  screw  with  20  threads 
per  inch.     Then  -=7X7  where    I  =  4    and  s  is  to  be  20. 

S  ta  tz 

ti^h       4        1 
Thus  T,  X  t*  =  20  =  5 

This  ratio  may  be  satisfied  by  using  the  following  gears  ti  =  20. 
ta  =  50,  tb  =  40  and  tz  =  80.  Only  the  one  idler  g  would  be 
used  to  give  the  right-hand  thread. 

2.  To  cut  a  standard  thread  on  a  2-in.  gas  pipe  in  the  lathe. 
The  proper  number  of  threads  here  would  be  llj^  per  inch  and 

hence  I  =  4,  s  =  llj^  and  r  X  .-  =  TTT?  =  oo'     This  could 

la  *2  11-72  46 

be  done  by  making  ti  =  40  and  tz  =  115,  and  tb  =  ta  both  acting 
as  one  idler. 

3.  If  it  were  required  to  cut  100  threads  per  inch  then  I  =  4, 

s  =  100  and  r  X  ^  =  TT^  =  ^>  which  may  be  divided  into  two 

ta         12 


parts,    thus    05  =  z  X  gr/»  so  that    making  ti  =  20,   ta  =  80, 


TRAINS  OF  GEARING 


119 


tz  =  75,  would  require  an  extra  gear  of  12  teeth  to  take  the 
place  of  6,  as  U  =  12. 

The  axle  holding  the  gears  a  and  6  may  be  changed  in  position 
so  as  to  make  these  gears  fit  in  all  cases  between  1  and  2.  The 
details  of  the  method  of  doing  this  are  omitted  in  the  drawing. 

In  order  to  show  how  much  gearing  has  been  used  in  the  mod- 
ern lathe,  the  details  of  the  gearing  for  the  headstock  of  the 
Hendey-Norton  lathe  are  given  in  Figs.  71,  72  and  73,  from 
figures  made  up  from  drawings  kindly  supplied  by  the  Hendey 
Machine  Co.,  Torrington,  Conn. 


FIG.  71. — Hendey-Norton  lathe. 

A  general  view  of  the  Hendey-Norton  lathe  is  shown  in 
Fig.  71,  and  a  detailed  drawing  in  Fig.  72  in  the  latter  of  which 
is  shown  a  belt  cone  with  four  pulleys,  P,  running  freely  on  the 
live  spindle  S.  Keyed  to  the  same  spindle  is  the  gear  shown  at 
Q,  and  secured  to  the  cone  P  is  the  pinion  T,  and  Q  and  T  mesh, 
when  required,  with  corresponding  gears  on  the  back  gearshaft 
R.  When  the  cone  is  driving  the  spindle  directly  the  pin  W, 
shown  in  the  gear  Q,  is  left  in  the  position  shown  in  the  drawing, 
thus  forcing  P  to  drive  the  spindle  through  Q,  but  when  the  back 
gear  is  to  be  used,  the  pin  W  is  drawn  back  out  of  contact  with 
the  cone  pulley,  the  shaft  R  is  revolved  by  means  of  the  handle 
A  so  as  to  throw  the  gears  on  it  into  mesh  with  T  and  Q  and 


120 


THE  THEORY  OF  MACHINES 


TRAINS  OF  GEARING 


121 


then  the  spindle  is  driven  from  the  cone,  through  T  and  the 
back  gears,  and  back  through  gear  Q  which  is  keyed  to  the 
spindle.  The  use  of  the  back  gears  enables  the  spindle  to  be 
run  at  a  much  slower  speed  than  the  cone  pulley. 

For  screw  cutting  the  back  gear  is  not  used,  but  a  train  of 
gears,  /,  F,  Z,  X,  C,  E,  F,  G,  H,  K,  L,  M,  N,  and  B,  and  an  idler 
(or  tumble  gear  as  it  is  called)  delivers  motion  from  the  live 
spindle  to  the  set  of  gears  A  on  the  lead  screw.  These  gears 
are  partly  shown  on  Fig.  72  and  partly  on  Fig.  73  which  latter 
is  diagrammatic.  Of  the  gears  mentioned  J  is  keyed  to  the  live 

Stud    D    is  Geared  to  Spindle    1  to  1 

Gear 
Stud  D-+ 

48  T 
Gear  C 


Position  of  Gears 
with  Handle  U 
*n*"l  Hole 


Position  of  Gears 
with  Handle  U 
in^2  Hole 


Position  of  Gears 
with  Handle   U 
in^3  Hole 


FIG.  73.  —  Gears  on  Hendey-Norton  lathe. 


spindle,  the  idler  Y  is  slipped  over  into  gear  when  a  screw  is  to 
be  cut  and  causes  the  gear  Z  to  turn  at  the  same  speed  as  the 
spindle.  The  gear  Z  gives  motion  to  the  stud  D  by  means  of 
the  bevel  gears  and  jaw  clutch  shown  below  gear  T,  and  the  sense 
of  motion  of  D  will  depend  on  whether  the  jaw  clutch  is  put  into 
contact  with  the  right-  or  left-hand  bevel  gear,  these  bevel  gears 
being  to  adapt  the  lathe  to  the  cutting  of  a  right-  or  left-hand 
thread.  Gear  X,  keyed  to  stud  D,  revolves  at  the  same  speed 
as  Z  and  therefore  as  the  spindle  S  and  in  the  same  or  opposite 
sense  to  it  according  to  the  position  of  the  clutch;  with  the  clutch 
to  the  left  both  would  turn  in  the  same  sense.  The  remainder 
of  the  train  of  gearing  is  indicated  clearly  on  Fig.  73,  and  it  is 
to  be  noted  that  where  several  of  these  gears  are  on  the  same 
shaft  they  are  fastened  together,  such  as  C  and  E,  G  and  F, 


122  THE  THEORY  OF  MACHINES 

L,  K,  H,  etc.  The  numbers  of  teeth  shown  in  the  various  gears 
correspond  to  those  used  in  the  16-in.  and  18-in.  lathes. 

The  handles  shown  control  the  gear  ratios;  thus  U  controls  the 
positions  of  the  gears  L,  K,  and  H  and  the  figure  shows  the  three 
possible  positions  provided  by  the  maker  and  corresponding  to 
the  three  holes  1,  2  and  3  in  Fig.  72.  The  gear  B  is  provided 
with  a  feather  running  in  a  long  key  seat  cut  in  the  shaft  shown, 
and  the  handle  V  is  arranged  so  as  to  control  the  horizontal 
position  of  the  gear  B  and  its  tumbler  gear;  that  is  the  handle 
V  enables  the  operator  to  bring  B  into  gear  with  any  of  the  12 
gears  on  the  lead  screw.  The  lead  screw  has  6  threads  per  inch. 

With  the  handle  U  in  No.  3  hole  and  the  handle  V  in  the  fourth 
hole  as  shown  in  the  right-hand  diagram  of  Fig.  73  the  ratio  is 

n2  _  I  _  h       tx       fc       /,        tK       fe 

"    WJL    ~    8    ~    tz    X    tC    X    /,    X    TH    X    tu  70 


1       48       68       34       34       70 

"   S  =  12  X  6  =  12  X  6  = 


or  the  lathe  would  be  set  to  cut  3J^  threads  per  inch. 

134.  Cutting  Special  Threads,  Etc. — When  odd  numbers  of 
threads  are  to  be  cut,  various  artifices  are  resorted  to  to  get  the 
required  gearing,  sometimes  approximations  only  being  employed. 
Thus  if  it  were  required  to  cut  threads  on  a  2-in.  gas  pipe,  which 
has  properly  llj^  threads  per  inch,  and  the  lathe  had  not  gears 
for  the  purpose,  it  might  be  possible  to  cut  11}^  threads  per  inch 
or  11%  threads  per  inch,  either  of  which  would  serve  such  a 
purpose  quite  well.  There  are  cases,  however,  where  exact 
threads  of  odd  pitches  must  be  cut  and  an  example  will  show  one 
method  of  getting  at  the  proper  gears. 

Let  it  be  required  to  cut  a  screw  with  an  exact  pitch  of  1  mm. 
(0.0393708  in.)  with  a  lathe  having  8  threads  per  inch  on  the 
leading  screw,  and  assume  te  =  th. 

A  convenient  means  of  working  out  this  problem  is  the  method 
of  continued  fractions. 

The  exact  value  of  the  ratio  ^  is 

_!  Y%  0.125 

R  ~  0.0393708  ~  0.0393708* 


TRAINS  OF  GEARING  123 

The  first  approximation  is 

1  1  68,876 

g  =  3,  the  real  value  being  ^  =  3  + 

The  second  is 

3  +  •=,  the  real  value  being  3  + 


5'  49,328 

5+  6^876 

and  in  this  way  the  third,  fourth,  fifth,  etc.,  approximations  are 
readily  found.     The  sixth  is 


1 


7_       127 
^40    "   40 

Thus  with  a  gear  of  40  teeth  at  1  or  ti  =  40  and  tz  =  127  on 
the  leading  screw,  and  an  idler  in  place  of  a  and  b  the  thread  could 
be  cut. 

0 125  127 

(Note  that  n  HOODOO  =  3.17494  while -^  =  3.175,  so  that  the 

U.Uoyo/Uo  4U 

arrangement  of  gears  would  give  the  result  with  great  accuracy.) 

Problems  of  this  nature  frequently  lend  themselves  to  this 
method  of  solution,  but  other  methods  are  sometimes  more  con- 
venient and  the  ingenuity  of  the  designer  will  lead  him  to  devise 
other  means. 

135.  Hunting  Tooth  Gears. — These  are  not  much  used  now 
but  were  formerly  employed  a  good  deal  by  millwrights  who 
thought  that  greater  evenness  of  wear  on  the  teeth  would  result 
when  a  given  pair  of  teeth  in  two  gears  came  into  contact  the 
least  number  of  times.  To  illustrate  this,  suppose  a  pair  of  gears 
had  80  teeth  each,  the  velocity  ratio  between  them  thus  being 
unity;  then  a  given  tooth  of  one  gear  would  come  into  contact 
with  a  given  tooth  of  the  other  gear  at  each  revolution  of  each 
gear,  but,  if  the  number  of  teeth  in  one  gear  were  increased  to  81 
then  the  velocity  ratio  is  nearly  the  same  as  before  and  yet  a 
given  pair  of  teeth  would  come  into  contact  only  after  80  revolu- 
tions of  one  of  the  gears  and  81  revolutions  of  the  other.  The 
odd  tooth  is  called  a  hunting  tooth.  Compare  the  case  where 
the  gears  have  40  teeth  and  41  teeth  with  the  case  cited. 


124 


THE  THEORY  OF  MACHINES 


EPICYCLIC  GEARING,  ALSO  CALLED  PLANETARY  GEARING 

136.  Epicyclic  Gearing. — An  epicyclic  train  has  been  defined 
at  the  beginning  of  the  chapter  as  one  in  which  one  of  the  gears 
in  the  train  is  held  stationary  or  is  prevented  from  turning, 
while  all  the  other  gears  revolve  relative  to  it.  The  frame 
carrying  the  revolving  gear  or  gears  must  also  revolve.  The  train 
is  called  epicyclic  because  a  point  on  the  revolving  gear  describes 
epicyclic  curves  on  the  fixed  one,  and  the  term  planetary  gearing 
appears  to  be  due  to  the  use  of  such  a  train  by  Watt  in  his  "sun 
and  planet"  motion  between  the  crankshaft  and  connecting  rod 
of  his  early  engines. 

An  epicyclic  train  of  gears  is  made  up  in  exactly  the  same 
way  as  an  ordinary  train  already  examined,  the  only  difference 


Epicyclic  trains. 


between  the  two  is  in  the  part  of  the  combination  that  is  fixed; 
in  the  ordinary  train  the  axles  on  which  the  gears  revolve  are 
fixed  in  space,  that  is,  the  frame  is  fixed  and  all  the  gears  revolve, 
whereas  in  the  epicyclic  train  one  of  the  gears  is  prevented  from 
turning  and  all  of  the  other  gears  and  the  frame  revolve.  This 
is  another  example  of  the  inversion  of  the  chain  explained  in 
Chapter  I. 

The  general  purpose  of  the  train  is  to  obtain  a  very  low 
velocity  ratio  without  the  use  of  a  large  number  of  gears;  thus  a 
ratio  of  Ko.ooo  may  ve-ry  smiply  be  obtained  with  four  gears,  the 
largest  of  which  contains  101  teeth.  It  also  has  other  applications. 
Any  number  of  wheels  may  be  used,  although  it  is  unusual  to 
employ  over  four. 

In  discussing  the  train,  the  term  "first  wheel"  will  correspond 
with  wheel  1  and  "  last  wheel "  with  wheel  2  in  the  train  shown  in 
Fig.  68,  and  it  will  always  be  the  first  wheel  which  is  prevented 


TRAINS  OF  GEARING  125 

from  turning.  The  ratio  of  the  train  is  the  number  of  turns  of 
the  last  wheel  for  each  revolution  of  the  frame. 

In  Fig.  74  two  forms  of  the  train,  each  containing  two  wheels, 
are  shown.  In  the  left-hand  figure,  wheel  1  is  fixed  in  space  and 
the  frame  F  and  wheel  2  revolve,  whereas  in  the  right-hand  figure 
the  wheel  1  is  fixed  only  in  direction,  being  connected  to  links  in 
such  a  way  that  the  arrow  shown  on  it  always  remains  vertical 
(a  construction  easily  effected  in  practice),  that  is  wheel  1  does 
not  revolve  on  its  axis,  and  the  frame  F  and  wheel  2  both  revolve 
about  the  center  B.  The  following  discussion  applies  to  either 
case. 

137.  Ratio  of  Epicyclic  Gearing. — Let  the  gears  1  and  2  con- 
tain ti  and  t%  teeth  respectively;  then  as  a  simple  train  the  ratio  is 

R  =  •-    and  is  negative,  since  the  first  and  last  wheels  turn  in 

the  opposite  sense.  The  method  of  obtaining  the  velocity  ratio 
of  the  corresponding  epicyclic  train  may  now  be  explained. 
Assume  first  that  the  frame  and  both  wheels  are  fastened  together 
as  one  body  and  the  whole  given  one  revolution  in  space;  then 
frame  F  turns  one  revolution,  and  also  the  gears  1  and  2  each 
turn  one  revolution  on  their  axes  (not  axles) .  But  in  the  epicyclic 
train  the  gear  1  must  not  turn  at  all,  hence  it  must  be  turned 
back  one  revolution  to  bring  it  back  to  its  original  state,  and  this 
will  cause  the  wheel  2  to  make  R  revolutions  in  the  same  sense  as 
before,  since  the  ratio  R  is  negative.  During  the  whole  operation 
gear  1  has  not  moved,  the  frame  F  has  made  1  revolution  and  the 
last  wheel  1  +  R  revolutions  in  the  same  sense,  hence  the  ratio 
of  the  train  is 

^  _  1  +  R  _  Revolutions  made  by  the  last  wheel. 
1  Revolutions  made  by  the  frame. 

A  study  of  the  problem  will  show  that  if  R  were  positive  then 
E  =  1  —  R  and  in  fact  the  correct  algebraic  "formula  is 

E  =  1  -  R 

and  in  substituting  in  this  formula  care  must  be  taken  to  attach 
to  R  the  correct  sign  which  belongs  to  it  in  connection  with  an 
ordinary  train. 

Owing  to  the  difficulty  presented  by  this  matter  the  following 
method  of  arriving  at  the  result  may  be  helpful,  and  in  this  case 
a  train  will  be  considered  where  R  is  positive,  i.e.,  there  are  an 
even  number  of  contacts. 


126  THE  THEORY  OF  MACHINES 

1.  Assume  the  frame  fixed  and  first  wheel  revolved  once;  then: 
Frame  makes  0  revolutions. 

First  wheel  makes  +  1  revolutions. 

Last  wheel  makes  +  R  revolutions. 

But  the  epicyclic  train  is  one  in  which  the  first  wheel  does  not 
revolve,  and  therefore  to  bring  it  back  to  rest  let  all  the  parts  be 
turned  one  revolution  in  opposite  sense  to  the  former  motion. 

2.  After  all  parts  have  been  turned  backward  one  revolution 
the  total  net  result  of  both  operations  is: 

Frame  has  made  0  —  1  revolutions. 

First  wheel  has  made   +1  —  1  revolutions. 

Last  wheel  has  made    -\-R-l  revolutions, 

which  has  brought  the  wheel  1  to  rest;  hence 

r>   _   -I 

E  =  -Q-^   j  -  =  1  —  R    as  before. 

138.  Examples.  —  The  following  examples  will  illustrate  the 
meaning  of  the  formula  and  the  application  of  the  train  in 
practice. 

1.  Let  wheel  1  have  60  teeth  and  wheel  2  have  59  teeth; 

60 
then  ti  =  60,  t%  —  59  and  therefore  R  =  —  -^ 

Hence,    E  =  l-R  =  l--         =  l+       =  or    the 


last  wheel  turns  in  the  same  sense  as  the  frame  and  at  about 
double  its  speed. 

2.  Suppose  now  that  an  idler  is  inserted  between  1  and  2, 

AO 
keeping  R  =        still,  but  making  it  positive. 


Thus,  the  wheel  2  turns  in  opposite  sense  to  the  frame  and  at 
3^9  its  speed. 

3.  If  in  example  (2)  wheels  1  and  2  are  interchanged,  then 

59 

R  =  —  and  is  positive,  so  that 
bU 

p       1       59  1 

"60=     ^60 

in  which  case  the  last  wheel  will  turn  in  the  same  sense  as  the 
frame  and  at  %o  of  its  velocity. 

4.  To  design  a  train  having  a  positive  ratio  of  in  QQQ>  that  is, 


TRAINS  OF  GEARING 


127 


one  in  which  the  last  wheel  turns  in  the  same  sense  as  the  frame 
and  at  TTT^T^  the  speed. 


10,000 


Here 


E  =  1  - R  = 


10,000 


or     R  =  1  - 


10,000 


/I       JLW 

\         1007  \ 


J_\  „-??-  v  *™. 

1007    "  100  A  100 


The  train  thus  consists  of  four  gears  1,  a,  b  and  2  and  the 
numbers  of  teeth  are  ti  =  99,  ta  =  100,  tb  =  101,  t2  =  100. 

In  practice  such  a  train  could  easily  be  reverted,  although  the 
numbers  of  teeth  are  not  exactly  suited  to  it,  and  would  work 
quite  smoothly.  The  train  is  frequently  made  up  in  the  form 


100 


FIG.  75. 

shown  in  Fig.  75  where  the  frame  takes  the  form  of  a  loose 
pulley  A,  carrying  axles  D  on  which  the  intermediate  gears  run 
and  the  99-toothed  gear  is  keyed  to  B,  which  is  also  keyed  to  the 
frame  C.  The  pulley  will  turn  10,000  times  for  each  revolution 
that  the  shaft  makes.  Should  the  gears  be  changed  around,  so 
that  the  100-tooth  wheel  is  fixed,  while  the  99-tooth  wheel  is  on 
the  shaft  and  gears  with  the  100-tooth  wheel  on  D,  then 

li- 

9,999' 

or  the  shaft  will  turn  slowly  in  opposite  sense  to  the  wheel  A 


P  _  1  v          _ 

"  101  X   99   " 


128  THE  THEORY  OF  MACHINES 

The  arrangement  sketched  in  Fig.  75,  in  a  slightly  modified 
form,  is  used  in  screw-cutting  machines,  but  with  a  much  larger 
value  of  E.  In  this  case  there  are  two  pulleys,  one  as  shown  at  A 
and  one  keyed  to  the  shaft,  while  the  gears  on  D  are  usually  re- 
placed by  a  broad  idler.  When  the  die  is  running  up  on  the  stock 
the  operation  is  slow  and  the  belt  is  on  the  pulley  A,  but  for  other 
operations  the  speed  is  much  increased  by  pushing  the  belt  over 
to  the  pulley  keyed  to  the  shaft,  the  gears  then  running  idly. 

5.  Watt's  Sun  and  Planet  Motion. — In  this  case  gear  2  was 
the  same  size  as  1  and  was  keyed  to  the  crankshaft,  while  the  gear 
1  was  secured  to  the  end  of  the  connecting  rod  and  a  link  kept  the 
two  gears  at  the  proper  distance  apart,  as  in  the  right-hand 
diagram  of  Fig.  74.  There  was,  of  course,  no  crank. 

Here  R  =     -  1  and  E  =  1  -  R  =  2. 

Therefore,  the  crankshaft  made  two  revolutions  for  each  two 
strokes  of  the  piston. 

139.  Machines  Using  Epicyclic  Gearing. — There  are  a  great 
many  illustrations  of  this  interesting  arrangement  and  space 
permits  the  introduction  of  only  a  very  few  of  these. 

(a)  The  Weston  Triplex  Pulley  Block. — A  form  of  this  block, 
which  contains  an  epicyclic  train  of  gearing,  is  shown  in  Fig.  76. 
The  frame  D  contains  bearings  which  carry  the  hoisting  sprocket 
F,  and  on  the  casting  carrying  the  hoisting  sprocket  are  axles 
each  carrying  a  pair  of  compound  gears  BC,  the  smaller  one  C 
gearing  with  an  annular  gear  made  in  the  frame  D,  while  the  other 
and  larger  gear  B  of  the  pair  meshes  with  a  pinion  A  on  the 
end  of  the  shaft  S  to  which  the  hand  sprocket  wheel  H  is  attached. 
When  a  workman  pulls  on  the  hand  sprocket  chain  he  revolves  H 
and  with  it  the  pinion  A  on  the  other  end  of  the  shaft,  which  in 
turn  sets  the  compound  gears  BC  in  motion.  As  one  of  these 
gears  meshes  with  the  fixed  annular  gear  on  the  frame  D  the  only 
motion  possible  is  for  the  axles  carrying  the  compound  gears  to 
revolve  and  thus  carry  with  them  the  hoisting  sprocket  F. 

In  the  one  ton  Weston  block  the  annular  wheel  has  49  teeth, 
the  gear  B  has  31  teeth,  C  has  12  teeth  and  the  pinion  A  has  13 
teeth.  For  the  train,  then,  evidently  R  is  negative  since  one 
wheel  is  annular  and 

Rm        ^  v?1-     -973 
"  12  X  13 

Hence 

E  =  1  -  R  =  1  -  (-  9.73)  =  10.73. 


TRAINS  OF  GEARING 


129 


So  that  there  must  be  10.73  turns  of  the  hand  wheel  to  cause 
one  turn  of  the  hoisting  wheel  F.  As  these  wheels  are  respec- 
tively 9%  in.  and  3>£  in.  diameter,  the  hand  chain  must  be 

9% 
moved   ~y/  X  10.73  =  33.2  ft.  to  cause  the  hoisting  chain  to 


FIG.  76. — Weston  triplex  block. 

move  1  ft.,  so  that  the  mechanical  advantage  is  33.2  to  1,  neglect- 
ing friction. 

(6)  Motor-driven  Portable  Drill. — A  form  of  air-driven  drill  is 
shown  at  Fig.  77  in  which  epicyclic  gearing  is  used.  This  drill 
is  made  by  the  Cleveland  Drill  Co.,  Cleveland,  and  the  figure 
shows  the  general  construction  of  the  drill  while  a  detail  is  also 
given  of  the  train  of  gearing  employed. 


130 


THE  THEORY  OF  MACHINES 


jGearfo* 
Operating 
!  Valve  j 
!  D  \ 


FIG.  77. — Cleveland  air-driven  portable  drill. 


TRAINS  OF  GEARING  131 

The  outer  casing  A  of  the  drill  is  held  from  revolving  by  means 
of  the  handles  H  and  Hf,  air  to  drive  the  motors  passing  in 
through  H.  Fastened  to  the  bottom  of  the  crankshaft  of  the  air 
motors  is  a  pinion  B  which  drives  a  gear  C  and  through  it  a 
second  pinion  D,  which  latter  revolves  at  the  same  speed  as  B 
and  operates  the  valve  for  the  motors.  The  gear  C  is  keyed 
to  a  shaft  which  has  another  gear  E  also  secured  to  it,  the  latter 
meshing  with  pinions  F  which  in  turn  mesh  with  the  internal 
gear  G  secured  to  the  frame  A.  The  gears  F  run  freely  on  shafts 
J  which  are  in  turn  secured  in  a  flange  on  the  socket  S,  which 
carries  the  drill. 

As  the  motors  operate  on  the  crankshaft  causing  it  to  revolve, 
the  pinion  B  turns  with  it  and  also  the  gear  C  and  with  it  the 
gear  E.  As  E  revolves  it  sets  the  gears  F  in  motion  and  as  these 
mesh  with  the  fixed  gear  G  the  only  thing  possible  is  for  the 
spindles  J  carrying  F  to  revolve  in  a  circle  about  the  center  of 
E  and  as  these  revolve  they  carry  with  them  the  drill  socket  S. 
In  one  of  these  drills  the  motor  runs  at  1,275  revolutions  per 
minute  and  the  numbers  of  teeth  in  the  gears  are  tB  =  14,  tc  =  70, 
tE  =  15,  tP  =  15  and  to  =  45.  The  speed  of  the  spindle  S 
will  then  be  1,275  +  {1  +  4%5  X  7%4}  =  80  revolutions  per 
minute. 

(c)  Automobile  Transmission. — Epicyclic  gearing  is  now 
commonly  used  in  automobiles  and  two  examples  are  given  here, 
in  concluding  the  chapter.  The  mechanism  shown  in  Fig.  78 ] 
is  used  in  Ford  cars  for  variable  speed  and  reversing.  The 
engine  flywheel  A  carries  three  axles  X  uniformly  spaced  around 
a  circle  and  each  carrying  loosely  three  gears  H,  G  and  K,  the 
three  gears  being  fastened  together  and  rotating  as  one  solid 
body.  Each  of  these  gears  meshes  with  another  one  which 
is  connected  by  a  sleeve  to  a  drum;  thus  H  gears  with  B  and 
through  it  to  the  drum  C  which  is  keyed  to  the  shaft  P 
passing  back  to  the  rear  axle.  Gear  G  meshes  with  the  gear  F. 
which  is  attached  to  the  drum  E,  while  K  meshes  with  the  gear 
J  on  the  drum  7.  The  disc  M  is  keyed  to  an  extension  of  the 
crankshaft  as  shown  and  carries  one  part  of  a  disc  clutch,  the 
other  part  of  which  is  carried  on  casting  C.  The  mechanism  for 
operating  this  clutch  is  not  shown  completely. 

The  driver  of  the  car  has  pedals  and  a  lever  under  his  control 

1  A  drawing  was  kindly  furnished  by  the  Ford  Motor  Co.,  Ford,  Ontario, 
for  the  purpose  of  this  cut. 


132 


THE  THEORY  OF  MACHINES 


and  it  is  beyond  the  present  purpose  to  discuss  the  action  of 
these  in  detail,  but  it  may  be  explained  that  these  control  band 
brakes,  one  about  the  drum  /,  another  about  the  drum  E  and 
a  third  about  the  drum  C,  and  in  addition  the  pedals  and  lever 
control  the  disc  clutch  between  C  and  M. 

In  this  mechanism  the  gears  have  the  following  numbers  of 
teeth:  ta  =  27  teeth,  to  =  33  teeth,  tK  =  24  teeth,  tB  =  27  teeth, 
tF  =  21  teeth  and  tj  =  30  teeth. 


FIG.  78. — Ford  transmission. 

Should  the  driver  wish  the  car  to  travel  at  maximum  speed  he 
throws  the  disc  clutch  into  action  which  connects  M  and  C  and 
thus  causes  the  power  shaft  P  to  run  at  the  same  speed  as  that 
of  the  engine  crankshaft.  If  he  wishes  to  run  at  slow  speed  he 
operates  the  pedal  which  applies  the  band  brake  to  the  drum  E, 
causing  the  latter  to  come  to  rest.  The  gears  F,  G,  H  and  B 
then  form  an  epicyclic  train  and  for  this 


21 

R  = 


X  ^  =  0.636  and  E  =  1  -  R  =  0.36. 


So  that  the  power  shaft  P  will  turn  forward,  making  36  revolu- 
tions for  each  100  made  by  the  crank. 


TRAINS  OF  GEARING 


133 


If  the  car  is  to  be  reversed,  drum  I  is  brought  to  rest  and  the 
train  consists  of  gears  /,  K,  H  and  B. 


Then 


OQ       07       5 
R  =        X  <     = 


E  =  I  -  R  =  - 


or  the  power  shaft  P  will  turn  in  opposite  sense  to  the  crank  and 
at  one-fourth  its  speed. 

The  brake  about  C  is  for  applying  the  brakes  to  the  car. 

(d)  Automobile  Differential  Gear.  —  The  final  illustration  is 
the  differential  used  on  the  rear  axle  of  Packard  cars.  This  is 


FIG.  79. — Automobile  differential  gear. 

shown  in  Fig.  79  which  is  from  a  Packard  pamphlet.  The  power 
shaft  P  attached  to  the  bevel  pinion  A  drives  the  bevel  gear  B 
which  has  its  axis  at  the  rear  axle  but  is  not  directly  connected 
thereto.  The  wheel  B  carries  in  its  web  bevel  pinions  C,  the  axles 
of  which  are  mounted  radially  in  B,  and  the  pinions  C  may  rotate 
freely  on  these  axles. 

The  rear  axle  S  is  divided  where  it  passes  B  and  on  one  part 
of  the  axle  there  is  a  bevel  gear  D  and  on  the  other  one  the  bevel 
gear  E  of  the  same  size  as  D.  When  the  car  is  running  on  a 
straight  smooth  road  the  two  wheels  and  therefore  the  two  parts 


134  THE  THEORY  OF  MACHINES 

S  of  the  rear  axle  run  at  the  same  speed  and  then  the  power  is 
transmitted  from  P  through  A  and  B  just  as  if  the  gears  C,  D 
and  E  formed  one  solid  body. 

In  turning  a  corner,  however,  the  rear  wheel  on  the  outer  part 
of  the  curve  runs  faster  than  the  inner  one,  that  is  D  and  E 
run  at  different  speeds  and  gear  C  rotates  slowly  on  its  axle. 
When  the  one  wheel  spins  in  the  mud,  and  the  other  one  remains 
stationary,  as  not  infrequently  happens  when  a  car  becomes 
stalled,  the  arrangement  acts  as  an  epicyclic  train  purely. 

QUESTIONS  ON  CHAPTER  VII 

1.  Find  the  velocity  ratio  for  a  train  of  gears  as  follows:  A  gear  of  30 
teeth  drives  one  of  24  teeth,  which  is  on  the  same  shaft  with  one  of  48  teeth; 
this  last  wheel  gears  with  a  pinion  of  16  teeth. 

2.  The  handle  of  a  winch  carries  two  pinions,  one  of  24  teeth,  the  other 
of  15  teeth.     The  former  may  mesh  with  a  60-tooth  gear  on  the  rope  drum 
or,  if  desired,  the  15-tooth  gear  may  mesh  with  one  of  56  teeth  on  the  same 
shaft  with  one  of  14  teeth,  this  latter  gear  also  meshing  with  the  gear  of  60 
teeth  on  the  drum.     Find  the  ratio  in  each  case. 

3.  Design  a  reverted  train  for  a  ratio  4  to  1,  the  largest  gear  to  be  not 
over  9  in.  diameter,  6  pitch. 

4.  A  gear  a  of  40  teeth  is  driven  from  a  pinion  c  of  15  teeth,  through  an 
idler  b  of  90  teeth.     Retaining  c  as  before,  also  the  positions  of  the  centers 
of  a  and  c,  it  is  required  to  drive  a  60  per  cent,  faster,  how  may  it  be  done? 

6.  A  car  is  to  be  driven  at  15  miles  per  hour  by  a  motor  running  at  1,200 
revolutions  per  minute.  The  car  wheels  are  12  in.  diameter  and  the  motor 
pinion  has  20  teeth,  driving  through  a  compound  train  to  the  axle;  design 
the  train. 

6.  In  a  simple  geared  lathe  the  lead  screw  has  5  threads  per  inch,  gear 
e  =  21  teeth,  h  =  42  teeth,  1  =  60  teeth  and  2  =  72  teeth;  find  the  thread 
cut  on  the  work. 

7.  It  is  desired  to  cut  a  worm  of  0.194  in.  pitch  with  a  lathe  as  shown  at 
Fig.  70,  using  these  change  gears;  find  the  gears  necessary. 

8.  Make  out  a  table  of  the  threads  that  can  be  cut  with  the  lathe  in  Fig. 
70  with  different  gears. 

9.  Make  a  similar  table  to  the  above  for  the   Hendey-Norton  lathe 
illustrated. 

10.  Design  an  auto  change-gear  box  of  the  selective  type,  with  three 
speeds  and  reverse,  ratios  1.8  and  3.2  with  %  pitch  stub  gears,  shaft  centers 
not  over  10  in. 

11.  A  motor  car  is  to  have  a  speed  of  45  miles  per  hour  maximum  with  an 
engine  speed  of  1,400  revolutions  per  minute.     What  reduction  will  be 
required  at  the  rear  axle  bevel  gears,  36-in.  tires?     At  the  same  engine 
speed  find  the  road  speed  at  reductions  of  4  and  2  respectively. 

12.  Design  the  gear  box  for  the  above  car  with  %  stub-tooth  gears,  shafts 
9  in.  centers. 


TRAINS  OF  GEARING  135 

13.  Prove  that  the  velocity  ratio  of  an  epicyclic  train  is  E  =  1  —  R, 

14.  Design  a  reverted  epicyclic  train  for  a  ratio  of  1  to  2,500. 

16.  In  a  train  of  gears  a  has  24  teeth,  and  meshes  with  a  12-tooth  pinion 
6  which  revolves  bodily  about  a,  and  6  also  meshes  with  an  internal  gear  c 
of  48  teeth.  Find  the  ratio  with  a  fixed  and  also  with  c  fixed. 


CHAPTER  VIII 
CAMS 

140.  Purpose  of  Cams. — In  many  classes  of  machinery  certain 
parts  have  to  move  in  a  non-uniform  and  more  or  less  irregular 
way.  For  example,  the  belt  shifter  of  a  planer  moves  in  an 
irregular  way,  during  the  greater  part  of  the  motion  of  the  planer 
table  it  remains  at  rest,  the  open  and  crossed  belts  driving  their 
respective  pulleys,  but  at  the  end  of  the  stroke  of  the  table  the 
belts  must  be  shifted  and  then  the  shifter  must  operate  quickly, 
moving  the  belts,  after  which  the  shifter  comes  again  to  rest  and 
remains  thus  until  the  planer  table  has  completed  its  next  stroke, 
when  the  shifter  operates  again.  The  valves  of  a  gas  engine 
afford  another  illustration,  for  these  must  be  quickly  opened  at 
the  proper  time,  held  open  and  then  again  quickly  closed.  The 
operation  of  the  needle  bar  of  a  sewing  machine  is  well  known 
and  the  irregular  way  in  which  it  moves  is  familiar  to  everyone. 

In  the  machines  just  described,  and  indeed  in  almost  all 
machines  in  which  this  class  of  motion  occurs,  the  part  which 
moves  irregularly  must  derive  its  motion  from  some  other  part 
of  the  machine  which  moves  regularly  and  uniformly.  Thus, 
in  the  planer  all  the  motions  of  the  machine  are  derived  from  the 
belts  which  always  run  at  steady  velocity;  further,  the  shaft 
operating  the  valves  of  a  gas  engine  runs  at  speed  proportional 
to  the  crankshaft  while  the  needle  bar  of  a  sewing  machine  is 
operated  from  a  shaft  turning  uniformly. 

The  problem  which  presents  itself  then  is  to  obtain  a  non- 
uniform  motion  in  one  part  of  a  machine  from  another  part  which 
has  a  uniform  motion,  and  it  is  evident  that  at  least  one  of  the 
links  connecting  these  two  parts  must  be  unsymmetrical  in 
shape,  and  the  whole  irregularity  is  usually  confined  to  one  part 
which  is  called  a  cam.  Thus  a  cam  may  be  defined  as  a  link 
of  a  machine,  which  has  generally  an  irregular  form  and  by 
means  of  which  the  uniform  motion  of  one  part  of  the  machine 
may  be  made  to  impart  a  desired  kind  of  non-uniform  motion 
to  another  part. 

136 


CAMS 


137 


Cams  are  of  many  different  forms  and  designs  depending 
upon  the  conditions  to  be  fulfilled.  Thus  in  the  sewing  machine 
the  cam  is  usually  a  slot  in  a  flat  plate  attached  to  the  needle 
bar,  in  the  gas  engine  the  cam  is  generally  a  non-circular  disc 
secured  to  a  shaft,  whereas  in  screw-cutting  machines  it  often 
takes  the  form  of  a  slot  running  across  the  face  of  a  cylinder,  and 
many  other  cases  might  be  cited,  the  variations  in  its  form  being 
very  great.  Some  forms  of  cams  are  shown  in  Fig.  80. 

Several  problems  connected  with  the  use  of  cams  will  explain 
their  application  and  method  of  design. 

141  Stamp-mill  Cam. — The  first  illustration  will  be  that  of 
the  stamp  mill  used  in  mining  districts  for  crushing  ores,  and  a 
general  view  of  such  a  mill  is  shown  in  Fig.  81.  Such  a  mill  con- 
sists essentially  of  a  number  of  stamps  A,  which  are  merely 


FIG.  80. — Forms  of  cams. 

heavy  pieces  of  metal,  and  during  the  operation  of  the  mill 
these  stamps  are  lifted  by  a  cam  to  a  desired  height,  and  then 
suddenly  allowed  to  drop  so  as  to  crush  the  ore  below  them.  The 
power  to  lift  the  stamps  is  supplied  through  a  shaft  B  which  is 
driven  at  constant  speed  by  a  belt,  and  as  no  work  is  done  by 
the  stamps  as  they  are  raised,  the  problem  is  to  design  a  cam 
which  will  lift  them  with  the  least  power  at  shaft  B,  and  after 
they  have  been  lifted  the  cam  passes  out  of  gear  and  the  weights 
drop  by  gravity  alone. 

Now,  it  may  be  readily  shown  that  the  force  required  to  move 
the  stamp  at  any  time  will  depend  upon  its  acceleration,  being 
least  when  the  acceleration  is  zero,  because  then  the  only  force 
necessary  is  that  which  must  overcome  the  weight  of  the  stamp 
alone,  no  force  being  required  to  accelerate  it.  Thus,  for  the 


138 


THE  THEORY  OF  MACHINES 


minimum  expenditure  of  energy,  the  stamp  must  be  lifted  at  a 
uniform  velocity,  and  the  problem,  therefore,  resolves  itself  into 
that  of  designing  a  cam  which  will  lift  the  stamp  A  at  uniform 
velocity. 

The  general  disposition  of  the  parts  involved,  is  shown  in  Fig. 
82,  where  B  represents  the  end  of  the  shaft  B  shown  in  Fig.  81, 


FIG.  81. — Stamp  mill. 

and  YF  represents  the  center  line  of  the  stamp  A,  which  does 
not  pass  through  the  center  of  B.  Let  the  vertical  shank  of  the 
stamp  have  a  collar  C  attached  to  it,  which  collar  comes  into 
direct  contact  with  the  cam  on  B\  then  the  part  C  is  usually 
called  the  follower,  being  the  part  of  the  machine  directly 
actuated  by  the  cam. 

It  will  be  further  assumed  that  the  stamp  is  to  be  raised  twice 


CAMS 


139 


for  each  revolution  of  the  shaft  B,  and  as  some  time  will  be  taken 
by  the  stamp  in  falling,  the  latter  must  be  raised  its  full  distance 
while  the  shaft  B  turns  through  less  than  180°.  Let  the  total 
lift  occur  while  B  turns  through  102°. 

Further,  let  the  total  lift  of  the  cam  be  h  ft.,  that  is,  let  the 
distance  0  —  6,  Fig.  82,  through  which  the  bottom  of  the  follower 
C  rises,  be  h  ft. 

The  construction  of  the  cam  may  now  be  begun.  Draw  BF 
perpendicular  to  YF  and  lay  off  the  angle  FBE  equal  to  102°. 
Next  divide  the  distance  0  —  6  =  h,  and  also  the  angle  FBE,  into 


E 


FIG.  82. — Stamp  mill  cam. 

any  convenient  number  of  equal  parts,  the  same  number  being 
used  in  each  case;  six  parts  have  been  used  in  the  drawing. 

Now  a  little  consideration  will  show  that  since  the  stamp  A 
and  also  the  shaft  B  are  to  move  at  uniform  speed,  the  distances 
0-1,1-2,  2-3,  etc.  and  also  the  angles  FBG,  GBH,  HBJ,  etc., 
must  each  be  passed  through  in  the  same  intervals  of  time  and  all 
these  intervals  of  time  must  be  equal.  With  center  B  and  radius 
BF  draw  a  circle  FGH  ...E  tangent  to  the  line  0  —  6  and  draw  GM, 
HN,  etc.,  tangent  to  this  circle  at  G,  H,  etc.  Now  while  the 
follower  is  being  lifted  from  0  to  1  the  shaft  B  is  revolved  through 
the  angle  FBG,  and  then  the  line  GM  will  be  vertical  and  must 
be  long  enough  to  reach  from  F  to  1  or  GM  should  equal  FL 
The  construction  is  completed  by  making  HN  =  F  —  2,  JP 


140 


THE  THEORY  OF  MACHINES 


=  F  —  3,  etc.,  and  in  this  way  locating  the  points  0,  M,  N,  P,  Q,  R 
and  S  and  a  smooth  curve  through  these  points  gives  the  face 
of  the  cam.  As  a  guide  in  drawing  the  curve  it  is  to  be  remem- 
bered that  MG,  NH,  etc.,  are  normals  to  it. 

A  hub  of  suitable  size  is  now  drawn  on  the  shaft,  the  dimen- 
sions of  the  hub  being  determined  from  the  principles  of  machine 
design,  and  curves  drawn  from  S  and  0  down  to  the  hub  complete 
the  design;  the  curve  from  S  must  be  so  drawn  that  the  follower 
will  not  strike  the  cam  while  falling. 

The  curve  OMN  .  .  .  S  is  clearly  an  involute  having  a  base 
circle  of  radius  BF,  or  the  curve  of  the  cam  is  that  which  would 
be  described  by  a  pencil  attached  to  a  cord  on  a  drum  of  radius 


FIG.  83. — Uniform  velocity  cam. 

BF,  the  cord  being  unwound  and  kept  taut.  The  dotted  line 
shows  the  other  half  of  the  cam. 

In  this  case  there  is  line  contact  between  the  cam  and  its 
follower,  that  is,  it  is  a  case  of  higher  pairing,  as  is  frequently, 
though  not  always,  the  case  with  cams. 

142.  Uniform  Velocity  Cam. — As  a  second  illustration,  take 
a  problem  similar  to  the  latter,  except  that  the  follower  is  to  have 
a  uniform  velocity  on  the  up  and  down  stroke  and  its  line  of 
motion  is  to  pass  through  the  shaft  B.  It  will  be  further  assumed 
that  a  complete  revolution  of  the  shaft  will  be  necessary  for  the 
up  and  down  motion  of  the  follower. 


CAMS  141 

Let  0  —  8,  Fig.  83  (a)  represent  the  travel  of  the  follower,  the 
latter  being  on  a  vertical  shaft,  with  a  roller  where  it  comes 
into  contact  with  the  cam.  Divide  0  —  8  into,  say,  eight  equal 
parts  as  shown,  further,  divide  the  angle  OBK  ( =  180°)  into  the 
same  number  of  equal  parts,  giving  the  angles  OBI',  l'B2', 
etc.  Now  since  the  shaft  B  turns  at  uniform  speed  the  center 
of  the  follower  is  at  1  when  Bl'  is  vertical  and  at  2  when  B2'  is 
vertical,  etc.,  hence  it  is  only  necessary  to  revolve  the  lengths 
Bl,  B2,  etc.,  about  B  till  they  coincide  with  the  lines  Bl',  B2', 
etc.,  respectively.  The  points  1',  2',  3',  will  be  obtained  on 
the  radial  lines  Bl',  B2r,  etc.,  as  the  distances  from  B  which  the 
center  of  the  follower  must  have  when  the  corresponding  line 
is  vertical.  With  centers  1',  2',  3',  etc.,  draw  circles  to  represent 
the  roller  and  the  heavy  line  shown  tangent  to  these  will  be 
the  proper  outline  for  one-half  of  the  cam,  the  other  half  being 
exactly  the  same  as  this  about  the  vertical  center  line.  Here 
again  there  is  higher  pairing  and  some  external  force  is  supposed 
to  keep  the  follower  always  in  contact  with  the  cam. 

A  double  cam  corresponding  to  the  one  above  described  is 
shown  at  Fig.  83  (6),  this  double  cam  making  the  follower  perform 
two  double  strokes  at  uniform  speed  for  each  revolution  of  the 
camshaft. 

143.  Cam  for  a  Shear. — The  problem  may  appear  in  many 
different  forms  and  the  case  now  under  consideration  assumes 
somewhat  different  data  from  the  former  two,  and  the  shear 
shown  in  Fig.  84  may  serve  as  a  good  illustration.  Suppose  it  is 
required  to  design  a  cam  for  this  shear;  it  would  usually  be  desir- 
able to  have  the  shear  remain  wide  open  during  about  one-half 
the  time  of  rotation  of  the  cam,  after  which  the  jaw  should  begin 
to  move  uniformly  down  in  cutting  the  plate  or  bar,  and  then 
again  drop  quickly  back  to  the  wide-open  position.  With  the 
shear  wide  open,  let  the  arm  be  in  the  position  A\B\  where  it  is 
to  remain  during  nearly  one-half  the  revolution  of  the  cam ;  then 
let  it  be  required  to  move  uniformly  from  AiBi  to  A  2^2  while 
the  cam  turns  through  120°,  after  which  it  must  drop  back  again 
very  quickly  to  AiBi. 

An  enlarged  drawing  of  the  right-hand  end  of  the  machine  is 
shown  at  Fig.  85,  the  same  letters  being  used  as  in  Fig.  84,  the 
lines  AiBi  and  A2B2  representing  the  extreme  positions  of  the 
arm  AB.  Draw  the  vertical  line  QBiB2  and  lay  off  the  angle 
BiQB'z  equal  to  120°;  this  then  is  the  angle  through  which  the 


142 


THE  THEORY  OF  MACHINES 


camshaft  must  turn  while  the  arm  is  moving  over  its  range  from 
AiBi  to  AzB2.  Now  divide  the  angle  BiOB2,  Fig.  84,  into  any 
number  of  equal  parts,  say  four,  by  the  lines  OC,  OD,  and  OE; 
these  lines  are  shown  on  Fig.  85.  Next,  divide  the  angle 


FIG.  84. 

BiOBz  into  the  same  number  of  parts  as  BiOB2,  that  is  four, 
by  the  lines  QC',  QD'  and  QEr. 

Now,  when  the  line  QBi  is  vertical  as  shown,  the  cam  must  be 
tangent  to  AiBi.     Next,  when  the  cam  turns  so  that  QCi  becomes 


FIG.  85. — Cam  for  shear. 

vertical,  the  arm  must  rise  to  C,  and  hence  in  this  position  the 
line  OC  must  be  tangent  to  the  cam  and  the  corresponding  out- 
line of  the  cam  may  thus  be  found.  Draw  the  arc  CC'  with 
center  Q,  and  through  C'  draw  a  line  making  the  same  angle 


CAMS 


143 


ai  with  QCf  that  OC  does  with  QC.  The  line  through  C'  is  a 
tangent  to  the  cam.  Similarly,  tangents  to  the  cam  through 
D',  E'  and  B'2  may  be  drawn  and  a  smooth  curve  drawn  in 
tangent  to  these  lines,  as  shown  in  Fig.  85. 

The  details  of  design  for  the  part  B'2  G  may  be  worked  out  if 
proper  data  are  given,  and  evidently  the  part  GFB  is  circular  and 
corresponds  with  the  wide-open  position  of  the  shear. 

144.  Gas-engine  Cam. — It  not  infrequently  happens  that  the 
follower  has  not  a  straight-line  motion  but  is  pivoted  at  some 
point  and  moves  in  the  arc  of  a  circle.  This  is  the  case  with 
some  gas  engines  and  an  outline  of  the  exhaust  cam,  camshaft 


FIG.  86. — Gas-engine  cams. 

lever  and  exhaust  valve  for  such  an  engine  is  shown  at  Fig.  86  (a), 
where  A  is  the  camshaft  and  B  is  the  pin  about  which  the  fol- 
lower swings.  This  presents  no  difficulties  not  already  discussed 
but  in  executing  such  a  design  care  must  be  taken  to  allow  for  the 
deviation  of  the  follower  from  a  radial  line,  and  if  this  is  not  done 
the  cam  will  not  do  the  work  for  which  it  was  intended. 

As  this  problem  occurs  commonly  in  practice,  it  may  be  as 
well  to  work  out  the  proper  form  of  cam.  The  real  difficulty 
is  not  in  making  the  design  of  the  cam,  but  in  choosing  the  correct 
data  and  in  determining  the  conditions  which  it  is  desired  to 
have  the  cam  fulfil.  A  very  great  deal  of  discussion  has  taken 


144  THE  THEORY  OF  MACHINES 

place  on  this  point,  and  as  the  matter  depends  primarily  on  the 
conditions  set  in  the  engine,  it  is  out  of  place  here  to  enter  into 
it  at  any  length.  Such  a  cam  should  open  and  close  the  valve 
at  the  right  instants  and  should  push  it  open  far  enough,  but  in 
addition  to  these  requirements  it  is  necessary  that  the  valve 
should  come  back  to  its  seat  quietly,  and  that  in  moving  it 
should  always  remain  in  contact  with  the  cam-actuated  operating 
lever.  Further,  there  should  be  no  undue  strain  at  any  part 
of  the  motion,  or  the  pressure  of  the  valve  on  the  lever  should  be 
as  nearly  uniform  and  as  low  as  possible,  during  its  entire 
motion. 

The  total  force  required  to  move  the  valve  at  any  instant  is 
that  necessary  to  overcome  the  gas  pressure  on  top  of  it,  plus 
that  necessary  to  overcome  the  spring,  plus  that  necessary  to  lift 
and  accelerate  the  valve  if  it  has  not  uniform  velocity.  The  gas 
pressure  is  great  just  at  the  moment  the  valve  is  opened  (the 
exhaust  valve  is  here  spoken  of)  and  immediately  falls  almost  to 
that  of  the  atmosphere,  while  the  spring  force  is  least  when  the 
valve  is  closed  and  most  when  the  valve  is  wide  open.  The 
weight  of  the  valve  is  constant  and  its  acceleration  is  entirely 
under  the  control  of  the  designer  of  the  cam.  Under  the  above 
circumstances  it  would  seem  that  the  acceleration  should  be  low 
at  the  moments  the  valve  is  opened  and  closed,  and  that  it 
might  be  increased  as  the  valve  is  raised,  although  the  increasing 
spring  pressure  would  prevent  undue  increase  in  acceleration. 

Again,  the  velocity  of  the  valve  at  the  moment  it  returns  to  its 
seat  must  be  low  or  there  will  be  a  good  deal  of  noise,  and 
the  cam  should  be  so  designed  that  the  valve  can  fall  rapidly 
enough  to  keep  the  follower  in  contact  with  the  cam,  or  the  noise 
will  be  objectionable.  The  general  conditions  should  then  be 
that  the  follower  should  start  with  a  small  acceleration  which 
may  be  increased  as  the  valve  opens  more,  and  that  it  must 
finish  its  stroke  at  comparatively  low  velocity. 

In  lieu  of  more  complete  data,  let  it  be  assumed  that  the 
valve  is  to  remain  open  for  120°  of  rotation  of  the  cam,  and  is  to 
close  at  low  velocity.  The  travel  of  the  valve  is  also  given  and 
it  is  to  remain  nearly  wide  open  during  20°  of  rotation.  It  will 
first  be  assumed  that  the  follower  moves  bira  radial  line  as  at  Fig. 
86  (6)  and  correction  made  later  for  the  deviation  due  to  the  arc. 

From  the  data  assumed  the  valve-  is  to  move  upward  for  50° 
of  rotation  of  the  cam  and  downward  during  the  same  interval, 


CAMS 


145 


and  as  the  camshaft  turns  at  constant  speed  each  degree  of  rota- 
tion represents  the  same  interval  of  time.  Let  the  acceleration 
be  as  shown  on  the  diagram  Fig.  87  (a)  on  a  base  representing 
degrees  of  camshaft  rotation,  which  is  also  a  time  base;  then  the 
assumed  form  of  acceleration  curve  will  mean  that  at  first  the 
acceleration  is  zero  but  that  this  rapidly  increases  during  the 
first  5°  of  rotation  to  its  maximum  va^e  at  which  it  remains  for 


120    Degrees 


also  Seconds 


Degrees 
also  Seconds 


Degrees 


also  Seconds 


FIG.  87. 


the  next  15°.  It  then  drops  rapidly  to  the  greatest  negative  value 
where  it  also  remains  constant  for  a  short  interval  and  then 
rapidly  returns  to  zero  at  which  it  remains  for  20°,  after  which 
the  process  is  repeated.  Such  a  curve  means  a  rapidly  increasing 
velocity  of  the  valve  to  its  maximum  value,  followed  by  a  rapid 
decrease  to  zero  velocity  corresponding  to  the  full  opening  of  the 
valve  and  in  which  position  the  valve  remains  at  rest  for  20°. 
The  valve  then  drops  rapidly,  reaching  its  seat  at  the  end  of  120° 
at  zero  velocity. 
10 


146 


&HE  THEORY  OF  MACHINES 


By  integrating  the  acceleration  curve  the  velocity  curve  is 
found  as  shown  at  (6)  Fig.  87,  and  making  a  second  integration 
gives  the  space  curve  shown  at  (c),  the  maximum  height  of  the 
space  curve  representing  the  assumed  lift  of  the  cam.  These 
curves  show  that  the  valve  starts  from  rest,  rises  and  finally  comes 
to  rest  at  maximum  opening;  it  then  comes  down  with  rapid 
acceleration  near  the  middle  of  its  stroke  and  comes  back  on  its 
seat  again  with  zero  velocity  and  acceleration  and  therefore 
without  noise. 


FIG.  88. — Gas-engine  cam. 

Having  now  obtained  the  space  curve  the  design  of  the  cam  is 
made  as  follows: 

In  Fig.  88,  let  A  represent  the  camshaft  and  the  circle  G 
represent  the  end  of  the  hub  of  the  cam,  the  diameter  of  which  is 
determined  by  considerations  of  strength.  There  is  always  a 
slight  clearance  left  between  the  hub  and  follower  so  that  the 
valve  may  be  sure  to  seat  properly  and  this  clearance  circle  is 
indicated  in  light  lines  by  C.  Lay  off  radii  (say)  10°  apart  as 
shown;  then  AD  and  AE,  120°  apart,  represent  the  angle  of  action 
of  the  cam.  Draw  a  circle  F  with  center  A  and  at  distance  from 
C  equal  to  the  radius  of  the  roller;  then  this  circle  F  is  the  base 


CAMS  147 

circle  from  which  the  displacements  shown  in  (c),  Fig.  87,  are  to  be 
laid  off,  and  this  is  now  done,  one  case  being  shown  to  indicate  the 
exact  method.  The  result  is  the  curve  shown  in  dotted  lines 
which  begins  and  ends  on  the  circle  F.  A  pair  of  compasses  are 
now  set  with  radius  equal  to  the  radius  of  the  roller  of  the  follower 
and  a  series  of  arcs  drawn,  as  shown,  all  having  centers  on  the 
dotted  curve.  The  solid  curve  drawn  tangent  to  these  arcs  is 
the  outline  of  the  cam  which  would  fulfil  the  desired  conditions 
provided  the  follower  moved  in  and  out  along  the  radial  line  from 
the  center  A  as  shown  at  Fig.  86  (6) . 

Should  the  follower  move  in  the  aro  of  a  circle  as  is  the  case 
in  Fig.  86  (a),  where  the  follower  moves  in  the  arc  of  a  circle 
described  about  B,  then  a  slight  modification  must  be  made  in 
laying  out  the  cam  although  the  curves  shown  at  (a),  (6)  and 
(c),  Fig.  87,  would  not  be  altered.  The  method  of  laying  out 
the  cam  from  Fig.  87  (c)  may  be  explained  as  follows:  From 
center  A  draw  a  circle  H  (not  shown  on  the  drawings)  of  radius 
AB  equal  to  the  distance  from  the  center  of  camshaft  to  the 
center  of  the  fulcrum  for  the  lever.  Then  set  a  pair  of  compasses 
with  a  radius  equal  to  the  distance  from  B  to  the  center  of  the 
follower,  and  with  centers  on  H  draw  arcs  of  circles  outward 
from  the  points  where  the  radial  lines  AD,  etc.,  intersect  the 
circle  F;  one  of  these  is  shown  in  Fig.  88.  All  distances  such  as 
a  are  then  laid  off  radially  from  F  but  so  that  their  termini  will 
be  on  the  arcs  just  described;  thus  the  point  K  will  be  moved 
over  to  L,  and  so  with  other  points.  The  rest  of  the  procedure 
is  as  in  the  former  case.  For  ordinary  proportions  the  two  cams 
will  be  nearly  alike. 

Should  the  follower  have  a  flat  end  without  a  roller,  as  is  often 
the  case,  then  the  circle  F  is  not  used  at  all  and  all  distances  such 
as  a  are  laid  off  on  radial  lines  from  the  circle  C  and  on  each 
radius  a  line  is  drawn  at  right  angles  to  such  radius  and  of  length 
to  represent  the  face  of  the  follower.  The  outline  of  the  cam 
is  then  made  tangent  to  these  latter  lines. 

Lack  of  space  prevents  further  discussion  of  this  very  interest- 
ing machine  part,  which  enters  so  commonly  and  in  such  a  great 
variety  of  forms  into  modern  machinery.  No  discussion  has 
been  given  of  cams  having  reciprocating  motion,  nor  of  those 
used  very  commonly  in  screw  machines,  in  which  bars  of  various 
shapes  are  secured  to  the  face  of  a  drum  and  form  a  cam  which 
may  be  easily  altered  to  suit  the  work  to  be  done  by  simply 


148  THE  THEORY  OF  MACHINES 

removing  one  bar  and  putting  another  of  different  shape  in  its 
place. 

After  a  careful  study  of  the  cases  worked  out,  however,  there 
should  be  no  great  difficulty  in  designing  a  cam  to  suit  almost 
any  required  set  of  conditions.  The  real  difficulty,  in  most  cases, 
is  in  selecting  the  conditions  which  the  cam  should  fulfil,  but 
once  these  are  selected  the  solution  may  be  made  as  explained. 

QUESTIONS  ON  CHAPTER  VIII 

1.  Design  a  disk  cam  for  a  stamp  mill,  for  a  flat-faced  follower,  the  line 
of  the  stamp  being  4  in.  from  the  camshaft.     The  stamp  is  to  be  lifted  9  in. 
at  a  uniform  rate. 

2.  Design  a  disk  cam  with  roller  follower  to  give  a  uniform  rate  of  rise  and 
fall  of  3  in.  per  revolution  to  a  spindle  the  axis  of  which  passes  through  the 
center  line  of  the  shaft. 

3.  Taking  the  proportions  of  the  parts  from  Fig.  84,  design  a  suitable 
cam  for  the  shear. 

4.  A  cam  is  required  for  a  1  in.  shaft  to  give  motion  to  a  roller  follower 
%  in.  diameter,  and  placed  on  an  arm  pivoted  6  in.  to  the  left  and  2  in. 
above  the  camshaft.     The  roller  (center)  is  to  remain  2  in.  above  the  cam- 
shaft center  for  200°  of  camshaft  rotation,  to  rise  %  in.  at  uniform  rate 
during  65°,  to  remain  stationary  during  the  next  30°,  and  then  to  fall  uni- 
formly to  its  original  position  during  the  next  65°.     Design  the  cam. 

5.  Design  a  cam  similar  to  Fig.  88  to  give  a  lift  of  0.375  in.  during  45°,  a 
full  open  period  of  valve  of  25°  and  a  closing  period  of  45°.     Base  radius 
of  cam  to  be  0.625  in.  and  roller  1  in.  diameter. 


CHAPTER  IX 
FORCES  ACTING  IN  MACHINES 

146.  External  Forces. — When  a  machine  is  performing  any 
useful  work,  or  even  when  it  is  at  rest  there  are  certain  forces 
acting  on  it  from  without,  such  as  the  steam  pressure  on  an 
engine  piston,  the  belt  pull  on  the  driving  pulley,  the  force  of 
gravity  due  to  the  weight  of  the  part,  the  pressure  of  the  water  on 
a  pump  plunger,  the  pressure  produced  by  the  stone  which  is 
being  crushed  in  a  stone  crusher,  etc.  These  forces  are  called 
external  because  they  are  not  due  to  the  motion  of  the  machine, 
but  to  outside  influence,  and  these  external  forces  are  trans- 
mitted from  link  to  link,  producing  pressures  at  the  bearings 
and  stresses  in  the  links  themselves.  In  problems  in  machine 
design  it  is  necessary  to  know  the  effect  of  the  external  forces  in 
producing  stresses  in  the  links,  and  further  what  the  stresses  are, 
and  what  forces  or  pressures  are  produced  at  the  bearings,  for  the 
dimensions  of  the  bearings  and  sliding  blocks  depend  to  a  very 
large  extent  upon  the  pressures  they  have  to  bear,  and  the  shape 
and  dimensions  of  the  links  are  determined  by  these  stresses. 

The  matter  of  determining  the  sizes  of  the  bearings  or  links 
does  not  belong  to  this  treatise,  but  it  is  in  place  here  to  deter- 
mine the  stresses  produced  and  leave  to  the  machine  designer 
the  work  of  making  the  links,  etc.,  of  proper  strength. 

In  most  machines  one  part  usually  travels  with  nearly  uniform 
motion,  such  as  an  engine  crankshaft,  or  the  belt  wheel  of  a 
shaper  or  planer,  many  of  the  other  parts  moving  at  variable 
rates  from  moment  to  moment.  If  the  links  move  with  variable 
speed  then  they  must  have  acceleration  and  a  force  must  be 
exerted  upon  the  link  to  produce  this.  This  is  a  very  important 
matter,  as  the  forces  required  to  accelerate  the  parts  of  a  machine 
are  often  very  great,  but  the  consideration  of  this  question  is  left 
to  a  later  chapter,  and  for  the  present  the  acceleration  of  the 
parts  will  be  neglected  and  a  mechanism  consisting  of  light, 
strong  parts,  which  require  no  force  to  accelerate  them,  will  be 
assumed  in  place  of  the  actual  one. 

149 


150  THE  THEORY  OF  MACHINES 

146.  Machine  is  Assumed  to  be  in  Equilibrium. — It  will  be 
further  assumed  that  at  any  instant  under  consideration,  the 
machine  is  in  equilibrium,  that  is,  no  matter  what  the  forces 
acting  are,  that  they  are  balanced  among  themselves,  or  the 
whole  machine  is  not  being  accelerated.     Thus,  in  case  of  a 
shaper,  certain  of  the  parts  are  undergoing  acceleration  at  various 
times  during  the  motion,  but  as  the  belt  wheel  makes  a  constant 
number  of  revolutions  per  minute  there  must  be  a  balance  be- 
tween the  resistance  due  to  the  cutting  and  friction  on  the  one 
hand  and  the  power  brought  in  by  the  belt  on  the  other.     In 
the  case   of   a    train  which  is   just   starting  up,  the  speed   is 
steadily     increasing     and    the     train     is     being     accelerated, 
which  simply  means  that  more  energy  is  being  supplied  through 
the  steam  than  is  being  used  up  by  the  train,  the  balance  of 
the  power  being  free  to  produce  the  acceleration,  and  the  forces 
acting  are  not  balanced.     When,  however,  the  train  is  up  to 
speed  and  running  at  a  uniform  rate  the  input  and  output  must 
be  equal,  or  the  locomotive  is  in  equilibrium,  the  forces  acting 
upon  it  being  balanced. 

147.  Nature  of  Problems  Presented. — The  most  general  form 
of  problem  of  this  kind  which  comes  up  in  practice  is  such  as 
this:  Given  the  force  required  to  crush  a  piece  of  rock,  what 
belt  pull  in  a  crusher  will  be  required  for  the  purpose?  or:  What 
turning  moment  will  be  required  on  the  driving  pulley  of  a 
punch  to  punch  a  given  hole  in  a  given  thickness  of  plate?  or: 
Given  an  indicator  diagram  for  a  steam  engine,  what  is  the  result- 
ing turning  moment  produced  on  a  crankshaft?,  etc.     Such  prob- 
lems may  be  solved  in  two  ways:  (a)  by  the  use  of  the  virtual 
center;  (6)  by  the  use  of  the  phorograph,  and  as  both  methods 
are  instructive  each  will  be  discussed  briefly. 

148.  Solutions  by  Use  of  Virtual  Centers. — This  method  de- 
pends upon  the  fundamental  principles  of  statics  and  the  general 
knowledge  of  the  virtual  center  discussed  in  Chapter  II.     The 
essential  principles  may  be  summed  up  in  the  following  three 
statements : 

If  a  set  of  forces  act  on  any  link  of  a  machine  then  there  will 
be  equilibrium,  provided: 

1.  That  the  resultant  of  the  forces  is  zero. 

2.  That  if  the  resultant  is  a  single  force  it  passes  through  a 
point  on  the  link  which  is  at  the  instant  at  rest.     Such  a  point 


FORCES  ACTING  IN  MACHINES  151 

may,  of  course,  be  permanently  fixed  or  at  rest,  or  only  tempora- 
rily so. 

3.  That  if  the  resultant  is  a  couple  the  link  has,  at  the  instant, 
a  motion  of  translation. 

The  first  statement  expresses  a  well-known  fact  and  requires 
no  explanation.  The  second  statement  is  rather  less  known  but 
it  simply  means  that  the  forces  will  be  in  equilibrium  if  their 
resultant  passes  through  a  point  which  is  at  rest  relative  to  the 
fixed  frame  of  the  machine.  No  force  acting  on  the  frame  of 
the  machine  can  disturb  its  equilibrium,  for  the  reason  that  the 
frame  is  assumed  fixed  and  if  the  frame  should  move  in  any 
case  where  it  was  supposed  to  remain  fixed,  it  would  simply 
mean  that  the  machine  had  been  damaged.  Further,  a  force 
passing  through  a  point  at  rest  is  incapable  of  producing 
motion. 

The  third  statement  is  a  necessary  consequence  of  the  second 
and  corresponds  to  it.  If  the  resultant  is  a  couple,  or  two  parallel 
forces,  then  both  forces  must  pass  through  a  point  at  rest,  which 
is  only  possible  if  the  point  is  at  an  infinite  distance,  or  the  link 
has  a  motion  about  a  point  infinitely  distantly  attached  to  the 
frame,  that  is  the  link  has  a  motion  of  translation. 

Let  a  set  of  forces  act  on  any  link  b  of  a  mechanism  in  which 
the  fixed  link  is  d;  then  the  only  point  on  b  even  temporarily  at 
rest  is  the  virtual  center  bd,  which  may  possibly  be  a  permanent 
center.  Then  the  forces  acting  can  be  in  equilibrium  only  if 
their  resultant  passes  through  bd,  and  if  the  resultant  is  a  couple 
both  forces  must  pass  through  bd,  which  must  therefore  be  at 
an  infinite  distance,  or  b  must  at  the  instant,  have  a  motion  of 
translation.  These  ppints  may  be  best  explained  by  some 
examples. 

149.  Examples. — 1.  Three  forces  PI,  P2  and  P3,  Fig.  89,  act  on 
the  link  6;  under  what  conditions  will  there  be  equilibrium? 
In  the  first  place  the  three  forces  must  all  pass  through  the  same 
point  A  on  the  link,  and  treating  P2  as  the  force'  balancing  PI 
and  P3,  then  in  addition  to  P2  passing  through  A  it  must  also 
pass  through  a  point  on  the  link  b  which  is  at  rest,  that  is  the 
point  bd.  This  fixes  the  direction  of  P2,  by  fixing  two  points  on 
it,  and  thus  the  directions  of  the  three  forces  PI,  P2  and  P3  are 
fixed  and  their  magnitudes  may  be  found  from  the  vector  triangle 
to  the  right  of  the  figure. 

2.  To  find  the  force  P2  acting  at  the  crankpin,  in  the  direction 


152 


THE  THEORY  OF  MACHINES 


of  the  connecting  rod,  Fig.  90,  which  will  balance  the  pressure  PI 
on  the  piston.  In  this  case  PI  and  P2  may  both  be  regarded  as 
forces  acting  on  the  two  ends  of  the  connecting  rod  and  the 
problem  is  thus  similar  to  the  last  one.  PI  and  P2  intersect  at  be; 
hence  their  resultant  P  must  pass  through  be  and  also  through 
the  only  point  on  b  at  rest,  that  is  bd,  which  fixes  the  position  and 


FIG.  89. 

direction  of  P  and  hence  the  relation  between  the  forces  may  be 
determined  from  the  vector  triangle.  This  enables  P2  to  be 
found  as  in  the  upper  right-hand  figure. 

The  moment   of  P2   on   the   crankshaft   is  P2  X  OD,   which 
may  readily  be  shown  by  geometry  to  be  equal  to  PI  X  0  —  ac 

that  is,  the  turning  effect  on  the  crankshaft 


snce  -~-  = 

1   2 


—  CIC 


FIG.  90. 

due  to  the  piston  pressure  PI  is  the  same  as  if  PI  was  transferred 
to  the  point  ac  on  the  crankshaft. 

Let  P3,  acting  normal  to  the  crank  a  through  the  crankpin, 
be  the  force  which  just  balances  PI;  it  is  required  to  find  PS. 
Now  P3  and  PI  intersect  at  H,  and  their  balancing  force  P' 
must  pass  through  H  and  through  bd  which  gives  the  direction 


FORCES  ACTING  IN  MACHINES 


153 


and  position  of  P'  and  the  vector  triangle  EFG  gives  P3  corre- 
sponding to  a  known  value  of  PI.  The  force  P3  is  called  the 
crank  effort  and  may  be  defined  as  the  force,  passing  through  the 
crankpin  and  normal  to  the  crank,  which  would  produce  the  same 
turning  moment  on  the  crank  that  the  piston  pressure  does. 
More  will  be  said  about  this  in  the  next  chapter. 

3.  Forces  PI  and  P2  act  on  a  pair  of  gear  wheels,  the  pitch 
circles  of  which  are  shown  in  Fig.  91;  it  is  required  to  find  the 
relation  between  them,  friction  of  the  teeth  being  neglected. 
Since  friction  is  not  considered,  the  direction  of  pressure  between 


FIG.  91. 

the  teeth  must  be  normal  to  them  at  their  point  of  contact,  and 
is  shown  at  P3  in  the  figure,  this  force  always  passing  through  the 
point  of  contact  of  the  teeth  and  always  through  the  pitch  point 
or  point  of  tangency  of  the  pitch  circles.1  For  the  involute 
system  of  teeth  P3  is  fixed  in  direction  and  coincides  with  the  line 
of  obliquity,  but  with  the  cycloidal  system  P3  becomes  more 
and  more  nearly  vertical  as  the  point  of  contact  approaches  the 
pitch  point.  Knowing  the  direction  of  P3  from  these  considera- 
tions, let  it  intersect  PI  and  P2  at  A  and  B  respectively.  On  the 
wheel  a  there  are  the  forces  PI,  P3  and  P,  the  latter  acting  through 
A  and  ad,  and  their  values  are  obtained  from  the  vector  triangle; 
and  on  b  the  forces  P3,  P%  and  P',  the  latter  acting  through  B 
1  For  a  complete  discussion  on  these  points  see  Chapter  V. 


154 


THE  THEORY  OF  MACHINES 


and  bd,  and  representing  the  bearing  pressure,  are  found  in  the 
same  way,  the  vector  polygon  on  the  left  giving  the  values  of  the 
several  forces  concerned  and  hence  Pz  if  PI  is  known. 

4.  The  last  example  taken  here  is  the  beam  engine  illustrated 
in  outline  in  Fig.  92,  and  the  problem  is  to  find  the  turning 
moment  produced  on  the  crankshaft  due  to  a  given  pressure  PI 
acting  on  the  walking  beam  from  the  piston.  Two  convenient 
methods  of  solution  are  available,  the  first  being  to  take  moments 
about  cd  and  in  this  way  to  find  the  force  Pz  acting  through  be 
which  is  the  equivalent  of  the  force  PI  at  C,  the  remainder  of  the 
problem  there  being  solved  as  in  Example  2. 


FIG.  92. 

It  is  more  general,  however,  and  usually  -simpler  to  determine 
the  equivalent  force  on  the  crankshaft  directly.  Select  any 
point  D  on  PI  and  resolve  PI  into  two  components,  one  P  passing 
through  the  only  point  on  the  beam  c  at  rest,  that  is  cd,  the 
other,  P3,  passing  through  the  common  point  ac  of  a  and  c.  The 
positions  of  P  and  P3  and  their  directions  are  known,  since  both 
pass  through  D  and  also  through  cd  and  ac  respectively;  hence  the 
vector  triangle  on  the  right  gives  the  forces  PS  and  P.  But 
Pa  acts  through  ac  on  a,  and  if  ad  —  E  =  h,  be  drawn  from  ad 
normal  to  Pa,  the  moment  of  PS  about  the  crankshaft  is  Pzh, 
which  therefore  balances  the  moment  produced  on  the  crankshaft 
by  the  pressure  PI  on  the  walking  beam.  The  magnitude  of  this 
moment  is,  of  course,  independent  of  the  position  of  the  point  D. 


FORCES  ACTING  IN  MACHINES  155 

150.  General  Formula. — The  general  formula  for  the  solu- 
tion of  all  such  problems  by  use  of  virtual  centers  is  as  follows : 
A  force  PI  acts  through  any  point  B  on  a  link  6;  it  is  required  to 
find  the  magnitude  of  a  force  P2,  of  known  direction  and  posi- 
tion, acting  on  a  link  e  which  will  exactly  balance  PI,  d  being 
the  fixed  link.     Find  the  centers  bd,  be  and  ed.    Join  B  to  be 
and  bd  and  resolve  PI  into  P3  in  the  direction  B  —  be  and  P4  in 
the  direction  B  —  bd;  then  the  moment  of  P3  about  de  must  be  the 
same  as  the  moment  of  P2  about  the  same  point  and  thus  P2 
is  known. 

151.  Solution  of  Such  Problems  by  the  Use  of  the  Phoro- 
graph. — In  solving  such  problems  as  are  now  under  considera- 
tion by  the  use  of  the  phorograph  the  matter  is  approached  from 
a  somewhat  different  standpoint,  and  as  there  is  frequent  occasion 
to  use  the  method  it  will  be  explained  in  some  detail. 

It  has  already  been  pointed  out  that  the  present  investigation 
deals  only  with  the  case  where  the  machine  is  in  equilibrium, 
or  where  it  is  not,  on  the  whole,  being  accelerated.  This  is 
always  the  case  where  the  energy  put  into  the  machine  per  second 
by  the  source  of  energy  is  equal  to  that  delivered  by  the 
machine,  for  example,  where  the  energy  per  second  delivered  by 
a  gas  engine  to  a  generator  is  equal  to  the  energy  delivered  to 
the  piston  by  the  explosion  of  the  gaseous  mixture,  friction 
being  neglected. 

Suppose  now  that  on  any  mechanism  there  is  a  set  of  forces 
PI,  P2,  PS,  etc.,  acting  on  various  links,  and  that  these  forces  are 
acting  through  points  having  the  respective  velocities  v\,  v%,  vs, 
etc.,  feet  per  second  in  the  directions  of  PI,  P2,  PS.  The  energy 
which  any  force  will  impart  to  the  mechanism  per  second  is 
proportional  to  the  magnitude  of  the  force  and  the  velocity 
with  which  it  moves  in  its  own  direction;  thus  if  a  force  of  20 
pd.  acts  at  a  point  moving  at  4  feet  per  second  in  the  direction 
of  the  force,  the  energy  imparted  by  the  latter  will  be  80  ft.-pd. 
per  second,  and  this  will  be  positive  or  negative  according  to 
whether  the  sense  of  force  and  velocity  are  the  same  or  different. 

The  above  forces  will  then  impart  respectively  P\v\,  P2^2, 
Ps^s,  etc.,  ft.-pd.  per  second  of  energy,  some  of  the  terms  being 
negative  frequently  and  the  direction  of  action  of  the  various 
forces  are  usually  different.  The  total  energy  given  to  the 
machine  per  second  is  Pi»i  +  P2v2  +  PsV3  +  etc.,  ft.-pd.  and  if 
this  total  sum  is  zero  there  will  be  equilibrium,  since  the  net 


156  THE  THEORY  OF  MACHINES 

energy  delivered  to  the  machine  is  zero.  This  leads  to  the  im- 
portant statement  that  if  in  the  machine  any  two  points  in  the 
same  or  different  links  have  identical  motions,  then,  as  far  as 
the  equilibrium  of  the  machine  is  concerned,  a  given  force  may 
be  applied  at  either  of  the  points  as  desired,  or  if  at  the  two  points 
forces  of  equal  magnitude  and  in  the  same  direction  but  opposite 
in  sense  are  applied  then  the  equilibrium  of  the  machine  will  be 
unaffected  by  these  two  forces,  for  the  product  Pv  will  be  the  same 
in  each  case,  but  opposite  in  sense,  and  the  sum  of  the  products 
Pv  will  be  zero. 

To  illustrate  these  points  further  let  any  two  points  B  and  E' 
in  the  same  or  different  links  in  the  mechanism  have  the  same 
motion,  and  let  any  force  P  act  through  B,  then  the  previous 
paragraph  asserts  that  without  affecting  the  conditions  of  equilib- 
rium in  any  way,  the  force  P  may  be  transferred  from  B  to  B', 
that  is  to  say  that  if  a  force  P  act  through  a  point  B  in  any  link, 
and  there  is  found  in  any  other  link  in  the  mechanism  a  point  B' 
with  the  same  motion  as  B,  the  force  P  will  produce  the  same  effect 
as  far  as  the  equilibrium  of  the  mechanism  is  concerned,  whether 
it  acts  at  B  or  B'. 

It  has  been  shown  in  Chapter  IV  that  to  each  point  in  a  median-* 
ism  there  may  be  found  a  point  called  its  image  on  a  selected 
link,  which  point  has  the  same  motion  as  the  point  under  dis- 
cussion, and  thus  it  is  possible  to  find  on  a  single  link  a  collec- 
tion of  points  having  the  same  motions  as  the  various  points 
of  application  of  the  acting  forces.  Without  affecting  the  con- 
ditions of  equilibrium,  any  force  may  be  moved  from  its  actual 
point  of  application  to  the  image  of  this  point,  and  thus  the 
whole  problem  be  reduced  to  the  condition  of  equilibrium  of 
a  set  of  forces  acting  on  a  single  link.  There  will  be  equilib- 
rium provided  the  sum  of  the  moments  of  the  forces  about  the 
center  of  rotation  relative  to  the  frame  is  zero. 

152.  Examples  Using  the  Phorograph. — As  this  matter  is 
somewhat  difficult  to  understand  it  may  best  be  explained  by  a 
few  practical  examples  in  which  the  application  is  given  and 
in  the  solution  of  the  problems  it  will  be  found  that  the  only  diffi- 
culty offered  is  in  the  finding  of  the  phorograph  of  the  mechanism, 
so  that  Chapter  IV  must  be  carefully  mastered  and  understood. 

1.  To  find  the  turning  effect  produced  on  the  crankshaft  of 
an  engine  due  to  the  weight  of  the  connecting  rod.  Let  Fig.  93 
represent  the  engine  mechanism,  with  connecting  rod  AB 


FORCES  ACTING  IN  MACHINES 


157 


having  a  weight  W  Ib.  and  with  its  center  of  gravity  at  G;  the 

weight  W  then  acts  vertically  downward  through  G.     Find  A', 

Bf  and  Gf,  the  images  of  A,  B  and  G  on  the  crank  OA ;  then  since, 

by  the  principle  of  the  phorograph,  the  motion  of  G'  is  identical 

with  that  of  G,  it  follows  that  Gf  must  have  exactly  the  same 

velocity  as  G,  that  is  to  say  energy  will  be  imparted  to  the 

mechanism  at  the  same  rate  per  second  by  the  force  W  acting 

at  G'  as  it  will  by  the   same 

force  acting  at  G,  so  that  the 

force  W  may  be  moved  to  G' 

without  affecting  the  conditions 

of   equilibrium,    and   this   has 

been    done  in  the  figures.     It  F      Q3 

must  not  be  supposed  that  W 

acts  both  at  G  and  G'  at  the  same  time ;  it  is  simply  transferred 

from  G  to  G'. 

Since  Gf  is  a  point  on  the  crankshaft,  the  moment  due  to  the 
weight  of  the  rod  is  Wh  ft.-pd.,  where  h  is  the  shortest  distance, 
in  feet,  from  0  to  the  direction  of  the  force  W. 

2.  A  shear  shown  in  Fig.  94  is  operated  by  a  cam  a  attached 
to  the  main  shaft  0,  the  shaft  being  driven  at  constant  speed  by 
a  belt  pulley.  Knowing  the  force  F  necessary  to  shear  the  bar 


\F        R 


K%^%^/^^%$%%^ 

FIG.  94.— Shear. 

at  8,  the  turning  moment  which  must  be  applied  at  the  camshaft 
0  is  required.  Let  P  be  the  point  on  the  cam  a  where  it  touches 
the  arm  b  at  Q,  then  the  motion  of  P  with  regard  to  Q  is  one  of 
sliding  along  the  common  tangent  at  P.  Choosing  a  as  the  link 
of  reference,  P'  will  lie  at  P,  Rr  at  0,  R'Q'  will  be  parallel  to 
RQ  and  Q'  will  lie  in  P'Q'  the  common  normal  to  the  surfaces 
at  P,  this  locates  Q'.  Having  now  two  points  on  b',  viz.,  R' 
and  Q',  complete  the  figure  by  drawing  from  Q'  the  line  Q'S' 
parallel  to  QS,  also  drawing  R'S'  parallel  to  RS  and  thus  locat- 
ing S'.  The  construction  lines  have  not  been  drawn  on  the 


158 


THE  THEORY  OF  MACHINES 


diagram.  The  figure  shows  the  whole  jaw  dotted  in,  although 
it  is  quite  unnecessary.  Having  now  found  Sf  a  point  on  a  with 
the  same  velocity  at  S  on  6,  the  force  F  may  be  transferred  to 
Sf  and  the  moment  F  X  h  of  F  about  0  is  the  moment  which  must 
be  produced  on  the  shaft  in  the  opposite  sense.  By  finding  the 


FIG.  95. — Rock  crusher. 

moment  in  a  number  of  positions  it  is  quite  easy  to  find  the 
necessary  power  to  be  delivered  by  the  belt  for  the  complete 
shearing  operation. 

3.  A  somewhat  more  complicated  machine  is  shown  in  Fig.  95, 
which  represents  a  belt-driven  rock  crusher  built  by  the  Fair- 


FORCES  ACTING  IN  MACHINES 


159 


banks-Morse  Co.,  the  lower  figure  having  been  redrawn  from 
their  catalogue,  and  the  upper  figure  shows  the  mechanism  on  a 
larger  scale. 

On  a  belt  wheel  shown,  the  belt  exerts  a  net  pull  Q  which 
causes  the  shaft  0,  having  the  eccentric  OA  attached  to  it,  to 
revolve.  The  shaft  H  carried  by  the  frame  has  the  arm  HB 
attached  to  it,  to  the  left-hand  end  of  which  is  a  roller  resting 
on  the  eccentric  OA.  The  crusher  jaw  is  pivoted  on  the  frame 
at  J  and  a  strong  link  CD  keeps  the  jaw  and  the  arm  HB  a  fixed 
distance  apart.  As  shaft  0  turns,  the  eccentric  imparts  a  motion 
to  the  arm  HB  which  in  turn  causes  the  jaw  to  have  a  pendulum 
motion  about  J  and  to  exert  a  pressure  P  on  a  stone  to  be  crushed. 
It  is  required  to  find  the  rela- 
tion between  belt  pull  Q  and  the 
pressure  P. 

Select  OA  as  the  link  of  ref- 
erence and  make  the  phorograph 
of  double  scale  as  in  Fig.  39,  mak- 
ing OA'  =  2  OA.  As  the  device 
simply  employs  two  chains 
similar  to  Fig.  32,  viz.,  OABCH 
and  JDCH,  the  images  of  all  the 
points  may  readily  be  found  and 
these  are  shown  on  the  figure. 
Then  P  is  transferred  from  G  to  Gr 
and  Q  from  E  to  Er  and  then  P 
and  Q  both  act  on  the  one  link 
and  hence  their  moments  must 
be  equal,  or  Q  X  OE'  =  mo- 
ment of  P  about  0,  from  which 
P  is  readily  found  for  a  given  value  of  Q. 

4.  The  application  to  a  governor1  is  shown  in  Fig.  96  which 
represents  one-half  of  a  Proell  governor,  and  it  is  required  to 
find  the  speed  of  the  vertical  spindle  which  will  hold  the  parts 
in  equilibrium  in  the  position  shown.  In  the  sketch  the  arm 
OA  is  pivoted  to  the  spindle  at  0  and  to  the  arm  BA  at  A,  the 
latter  arm  carrying  the  ball  C  on  an  extension  of  it  and  being 
attached  to  the  central  weight  W  at  B.  The  weight  of  each 

revolving  ball  at  C  is  -     Ib.  and  of  the  central  weight  is  W  Ib. 


FIG.  96. — Proell  governor. 


A  complete  discussion  of  governors  is  given  in  Chapter  XII. 


160  THE  THEORY  OF  MACHINES 

Treating  OA  as  the  link  of  reference  #nd  G  as  the  center  of  it, 

W 

find  the  images  of  A'  at  A  and  also  B'  and  C",  then  transfer  -~- 

jj 

(one-half  the  central  weight  acts  on  each  side)  to  Bf  and  ^  to  C', 

£t 

and  if  it  is  desired  to  allow  for  the  weights  wa  and  Wb  of  the  arms 
OA  and  A  B  the  centers  of  gravity  G  and  #  of  the  latter  are 
found  and  also  their  images  G'  and  Hf,  then  Wb  is  transferred  to 
H',  but  as  (r'  is  at  G,  wa  is  not  moved.  If  the  balls  revolve 
with  linear  velocity  v  ft.  per  second  in  a  circle  of  radius  r  ft., 

w       v^ 
then  the  centrifugal  force  acting  on  each  ball  will  be  P  —  ~~  X  — 

pds.  in  the  horizontal  direction,  and  this  force  P  is  trans- 
ferred to  C'.  Let  the  shortest  distances  from  the  vertical  line 
through  0  to  Bf,  C',  G'  and  H'  be  hi,  h2  7&3  and  h*  respectively, 
and  let  the  vertical  distance  from  C"  to  OB'  be  h$,  then  for  equili- 
brium of  the  parts  (neglecting  friction),  taking  moments  about 
0. 

~2'hi  +  -^-h2  +  wa  h3  +  Wb  h4  =  ^-  X  -  X  hb 

which  enables  the  velocity  v  necessary  to  hold  the  governor  in 
equilibrium  in  any  given  position  to  be  found,  and  from  this 
the  speed  of  the  spindle  may  be  computed. 

5.  The  chapter  will  be  concluded  by  showing  two  very  interest- 
ing applications  to  riveters  of  toggle-joint  construction.  The 
first  one  is  shown  in  Fig.  97,  the  drawing  on  the  left  showing  the 
construction  of  the  machine,  while  on  the  right  is  shown  the 
mechanism  involved  and  the  solution  for  finding  the  pressure  P 
at  the  piston  necessary  to  exert  a  desired  rivet  pressure  R. 
The  frame  d  carries  the  cylinder  g,  with  piston  /,  which  is  con- 
nected to  the  rod  e  by  the  pin  C.  At  the  other  end  of  e  is  a  pin 
A  which  connects  e  with  two  links  a  and  6,  the  former  of  which  is 
pivoted  to  the  frame  at  0.  The  link  b  is  pivoted  at  B  to  the  slide 
c  which  produces  the  pressure  on  the  rivet. 

Select  a  as  the  primary  link  because  it  is  the  only  one  having 
a  fixed  point;  then  A'  is  at  A}  and  since  B  has  vertical  motion, 
therefore  B'  will  lie  on  a  horizontal  line  through  0  and  also  on  a 
line  through  A'  in  the  direction  of  b,  that  is,  on  b  produced  so 
that  B'  is  found.  C'  lies  on  a  line  through  0  normal  to  the  direc- 
tion of  motion  of  C,  that  is  to  the  axis  of  the  cylinder  g,  and  since 


FORCES  ACTING  IN  MACHINES 


161 


it  also  lies  on  a  line  through  A'  parallel  to  e,  therefore  C"  is 
found. 

By  transferring  P  to  C'  and  R  to  B'  as  shown  dotted,  the  rela- 
tions between  the  forces  P  and  R  are  easily  found,  since  their 
moments  about  0  must  be  equal,  that  is,  P  X  OCf  =  R  X  OB'. 

By  comparing  the  first  and  later  positions  in  this  and  the 
following  figures  the  rapid  increase  in  the  mechanical  advantage 
of  the  mechanism,  as  the  piston  advances,  will  be  quite  evident. 

6.  Another  form  of  riveter  is  shown  at  Fig.  98  and  the  solution 
for  finding  the  rivet  pressure  R  corresponding  to  a  given  piston 
pressure  P  is  shown  along  with  the  mechanism  on  the  right  in 


Later  Position 


FIG.  97. — Riveter. 


two  positions.  The  proportions  in  the  mechanism  have  been 
altered  to  make  the  illustration  more  clear.  The  loose  link  b 
contains  four  pivots,  C}  B,  A,  F;  C  being  jointed  to  the  frame 
at  D  by  the  link  e;  B  having  a  connection  to  the  link  c,  which  link 
is  also  connected  at  E  to  the  sliding  block  e  acting  directly  on  the 
rivet.  A  is  connected  to  the  frame  at  0  by  means  of  the  link  a, 
and  F  is  connected  to  the  piston  g  at  G  by  means  of  the  link  /. 
Either  links  a  or  e  may  be  used  as  the  link  of  reference,  as 
each  has  a  fixed  center,  the  link  a  having  been  chosen.  The 
images  are  found  in  the  following  order:  C'  is  on  A'C'  and  on  D'C' 
parallel  to  DC;  B'  is  next  found  by  proportion,  as  is  also  F'  and 
thus  the  image  of  the  whole  link  b.  Next  E''is  on  B'E',  parallel 
to  BE  and  on  O'E'  drawn  perpendicular  to  the  motion  of  the 
slide  e,  while  G'  is  on  a  line  through  0  perpendicular  to  the  motion 

of  the  piston  g  and  is  also  on  the  line  F'G'  parallel  to  FG. 
n 


162 


THE  THEORY  OF  MACHINES 


Transfer  the  force  P  from  G  to  G',  and  the  force  R  from  E  to 
E',  and  then  the  moment  about  0  of  R  through  Ef  must  equal  the 
moment  of  P  through  Gf,  that  is,  R  X  OEr  =  P  X  OG'  from 
which  the  relation  between  R  and  P  is  computed  and  this  may  be 
done  for  all  the  different  positions  of  the  piston  g. 


D    Later  Position 


FIG.  98.— Riveter. 


QUESTIONS  ON  CHAPTER  IX 


1.  Why  are  external  forces  so  named? 
the  machine? 


What  effects  do  they  produce  in 


Solve  the  following  by  virtual  centers: 

2.  Determine  the  crank  effort  and  torque  when  the  crank  angle  is  45° 
in  an  8  in.  by  10-in.  engine  with  rod  20  in.  long,  the  steam  pressure  being  40 
pds.  per  square  inch. 

3.  In  a  pair  of  gears  15  in.  and  12  in.  diameter  respectively  the  direction 
of  pressure  between  the  teeth  is  at  75^°  to  the  line  of  centers,  which  is  hori- 
zontal.    On  the  large  gear  there  is  a  pressure  of  200  pds.  sloping  upward  at 
8°  and  its  line  of  action  is  3  in.  from  the  gear  center.     On  the  smaller  gear  is  a 
force  P  acting  downward  at  10°  and  to  the  left,  its  line  of  action  being  4  in. 
from  the  gear  center.     Find  P. 

4.  In  a  mechanism  like  Fig.  37  a  =  15  in.,  b  =  24  in.,  d  —  4  in.  and  e  =  60 
in.  and  the  link  a  is  driven  by  a  belt  on  a  10-in.  pulley  sloping  upward  at  60°. 
Find  the  relation  between  the  net  belt  pull  and  the  pressure  on  /  when  a  is 
at  45°. 

6.  The  connecting  rod  of  a  10  in.  by  12-in.  engine  is  30  in.  long  and  weighs 
30  lb.,  its  center  of  gravity  being  12  in.  from  the  crankpin.  What  turning 
effect  does  the  rod's  weight  produce  for  a  30°  crank  angle? 


FORCES  ACTING  IN  MACHINES  163 

Solve  the  following  by  the  phorograph: 

6.  In  a  crusher  like  Fig.  95,  using  the  same  proportions  as  are  there  given, 
find  the  ratio  of  the  belt  pull  to  the  jaw  pressure  and  plot  this  ratio  for  the 
complete  revolution  of  the  belt-wheel. 

7.  In  the   Gnome  motor,  Fig.  178,  with  a  fixed  link  2  in.  long,  and  the 
others   in   proportion  from   the  figure,   find  the  turning  moment  on  the 
cylinder  due  to  a  given  cylinder  pressure. 


CHAPTER  X 
CRANK-EFFORT    AND    TURNING-MOMENT    DIAGRAMS 

153.  Variations  in  Available  Energy . — In  the  case  of  all  engines, 
whether  driven  by  steam,  gas  or  liquid,  the  working  fluid  delivers 
its  energy  to  one  part  of  the  machine,  conveniently  called  the 
piston,  and  it  is  the  purpose  of  the  machine  to  convert  the  energy 
so  received  into  some  useful  form  and  deliver  it  at  the  shaft  to 
some  external  machine.  Pumps  and  compressors  work  in  exactly 
the  opposite  way,  the  energy  being  delivered  to  them  through 
the  crankshaft,  and  it  is  their  function  to  transfer  the  energy 
so  received  to  the  water  or  gas  and  to  deliver  the  fluid  in  some 
desired  state. 

In  the  case  of  machines  having  pure  rotary  motion,  such  as 
steam  and  water  turbines,  turbine  pumps,  turbo-compressors, 
etc.,  there  is  always  an  exact  balance  between  the  energy  supplied 
and  that  delivered,  and  the  input  to  and  output  from  the 
machine  is  constant  from  instant  to  instant.  Where  reciprocat- 
ing machines,  having  pistons,  are  used  the  case  becomes  somewhat 
different,  and  in  general,  the  energy  going  to  or  from  the  piston 
at  one  instant  differs  from  that  at  the  next  instant  and  so  on. 
This  of  necessity  causes  the  energy  available  at  the  crankshaft 
to  vary  from  time  to  time  and  it  is  essential  that  this  latter  energy 
be  known  for  any  machine  under  working  conditions. 

These  facts  are  comparatively  well  known  among  engineers. 
Steam  turbines  are  never  made  with  flywheels  because  of  the 
steadiness  of  motion  resulting  from  the  manner  of  transforming 
the  energy  received  from  the  steam.  On  the  other  hand,  recipro- 
cating steam  engines  are  always  constructed  with  a  flywheel, 
or  what  corresponds  to  one,  which  will  produce  a  steadying 
effect  and  the  size  of  the  wheel  depends  on  the  type  of  engine  very 
largely.  Thus,  a  single-cylinder  engine  would  have  a  heavy 
wheel,  a  tandem  compound  engine  would  also  have  a  heavy 
one,  while  for  a  cross-compound  engine  for  the  same  purpose  the 
flywheel  could  be  much  smaller  and  lighter. 

Again,  a  single-cylinder,  four-cycle,  single-acting  gas  engine 
would  have  a  much  larger  wheel  than  any  form  of  steam  engine, 

164 


CRANK-EFFORT  AND  TURNING-MOMENT       165 


and  the  flywheel  size  would  diminish  as  the  number  of  cylinders 
increased,  or  as  the  engine  was  made  double-acting  or  made  to 
run  on  the  two-cycle  principle,  simply  because  the  input  to  the 
pistons  becomes  more  constant  from  instant  to  instant,  and  the 
energy  delivered  by  the  fluid  becomes  more  steady. 

In  order  that  the  engineer  may  understand  the  causes  of  these 
differences,  and  may  know  how  the  machines  can  best  be  designed, 
the  matter  will  here  be  dealt  with  in  detail  and  the  first  case 
examined  will  be  the  steam  engine. 

154.  Torque. — An  outline  of  a  steam  engine  is  shown  in  Fig. 
99,  and  at  the  instant  that  the  machine  is  in  this  position  let 


Jl 


i , , ,, ,,  1 1 ,,~ ,  rr?u.       rr-i 
////////////////////I 

Ir 


FIG.  99. 

the  steam  produce  a  pressure  P  on  the  piston  as  indicated  (the 
method  of  arriving  at  P  will  be  explained  later),  then  it  is  re- 
quired to  find  the  turning  moment  produced  by  this  pressure  on 
the  crankshaft.  It  is  assumed  that  the  force  P  acts  through 
the  center  of  the  wristpin  B. 

Construct  the  phorograph  of  the  machine  and  find  the  image 
B'  of  B  by  the  principles  laid  down  in  Chapter  IV.  Now  in 
Chapter  IX  it  is  shown  that,  for  the  purposes  of  determining  the 
equilibrium  of  a  machine,  any  acting  force  may  be  transferred 
from  its  actual  point  of  application  to  the  image  of  its  point  of 
application.  Hence,  the  force  P  acting  through  B  will  produce 
the  same  effect  as  if  this  force  were  transferred  to  B'  on  the 
crankdisc,  so  that  the  turning  moment  produced  on  the  crank- 
disc  and  shaft  is  P  X  OB'  ft.-pds.,/ind  this  turning  moment  will 
be  called  the  torque  T. 

Thus  T  =  P  X  OB'  ft.-pds.  where  OB'  is  measured  in  feet. 

155.  Crank  Effort. — Now  let  the  torque  T  be  divided  by  the 
length  a  of  the  crank  in  feet,  then  since  a  is  constant  for  all  crank 
positions,  the  quantity  so  obtained  is  a  force  which  is  proportional 
to  the  torque  T  produced  by  the  steam  on  the  crankshaft.  This 


166  THE  THEORY  OF  MACHINES 

force  is  usually  termed  the  crank  effort  and  may  be  defined  as 
the  force  which  if  acting  through  the  crankpin  at  right  angles 
to  the  crank  would  produce  the  same  turning  effect  that  the 
actual  steam  pressure  does  (see  Sec.  149  (2)). 
Let  E  denote  the  crank  effort;  then 

EXa  =  T  =  PxOB'  ft.-pds. 


It  is  evident  that  the  turning  moment  produced  on  the  crank- 
shaft by  the  steam  may  be  represented  by  either  the  torque  T 
ft.-pds.  or  by  the  crank  effort  E  pds.,  since  these  two  always  bear  a 
constant  relation  to  one  another.  For  this  reason,  crank  efforts 
and  torques  are  very  frequently  confused,  but  it  must  be  re- 
membered that  they  are  different  and  measured  in  different 
units,  and  the  one  always  bears  a  definite  relation  to  the  other. 
The  graphical  solution  for  finding  the  effort  E  corresponding 
to  the  pressure  P  is  shown  in  Fig.  99.  It  is  only  necessary  to  lay 
off  OH  along  a  to  represent  P  on  any  convenient  scale,  and  to 
draw  HK  parallel  to  A  'B'  ,  and  then  the  length  OK  will  represent 
E  on  the  same  scale  that  OH  represents  P.  The  proof  is  simple. 
Since  the  triangles  OB'  A'  and  OKH  are  similar,  it  is  evident  that: 

OK       OB'       OB'       E   . 

OH  =  OA    =  ~a~  =  P  smce      X  a  =  P  X  OB  ' 

156.  Crank  Effort  and  Torque  Diagrams.  —  Having  now  shown 
how  to  obtain  the  crank  effort  and  torque,  it  will  be  well  to  plot 
a  diagram  showing  the  value  of  these  for  each  position  of  the 
crank  during  its  revolution.     Such  a  diagram  is  called  a  crank- 
effort  diagram  or  a  torque  diagram.     In  drawing  these  diagrams 
the  usual  method  is  to  use  a  straight  base  for  crank  positions, 
the  length  of  the  base  being  equal  to  that  of  the  circumference 
of  the  crankpin  circle. 

157.  Example.  —  Steam  Engine.  —  The  method  of  plotting  such 
curve  from  the  indicator  diagrams  of  a  steam  engine  is  given  in 
detail  so  that  it  may  be  quite  clear. 

Let  the  indicator  diagrams  be  drawn  as  shown  in  Fig.  100,  an 
outline  of  the  engine  being  shown  in  the  same  figure,  and  the 
crank  efforts  and  torques  will  be  plotted  for  24  equidistant  posi- 
tions of  the  crankpin,  that  is  for  each  15°  of  crank  angle.  The 


CRANK-EFFORT  AND  TURNING-MOMENT        167 

straight  line  OX  in  Fig.  101  is  to  be  used  as  the  base  of  the  new 
diagram,  and  is  made  equal  in  length  to  the  crankpin  circle, 
being  divided  into  24  equal  parts.  The  corresponding  numbers 
in  the  two  figures  refer  to  the  same  positions. 

The  vertical  line  OL  through  0  will  serve  as  the  axis  for  torques 
and  crank  efforts,  but,  of  course,  the  scale  for  crank  efforts  must 
be  different  from  that  for  torques. 

Let  A i  and  A2  represent  respectively  the  areas  in  square 
inches  of  the  head  and  crank  ends  of  the  piston,  the  difference 
between  the  two  being  due  to  the  area  of  the  piston  rod;  the 
stroke  of  the  piston  is  L  ft. 

Suppose  the  indicator  diagrams  to  be  drawn  to  scale  s,  by 
which  is  meant  that  such  a  spring  was  used  in  the  indicator  that 


FIG.  100. 

1  in.  in  height  on  the  diagram  represents  s  pds.  per  square  inch 
pressure  on  the  engine  piston;  thus  if  s  =  60  then  each  inch  in 
height  on  the  diagram  represents  a  pressure  of  60  pds. per  square 
inch  on  the  piston.  The  lengths  of  the  head-  and  crank-end 
diagrams  are  assumed  as  li  and  Z2  in.  (usually  li  =  Z2)  and  these 
lengths  rarely  exceed  4  in.  irrespective  of  the  size  of  the  engine. 
Now  place  the  diagrams  above  the  cylinder  as  in  Fig.  100 
with  the  atmospheric  lines  parallel  to  the  line  of  motion  of  the 
piston.  The  two  diagrams  have  been  separated  here  for  the 
sake  of  clearness,  although  often  they  are  superimposed  with 
the  atmospheric  lines  coinciding.  Further,  the  indicator  dia- 
gram lengths  have  been  adjusted  to  suit  the  length  representing 
the  travel  of  the  piston.  While  this  is  not  necessary,  it  will  fre- 
quently be  found  convenient,  but  all  that  is  really  required  is  to 


168  THE  THEORY  OF  MACHINES 

draw  on  the  diagrams  a  series  of  vertical  lines  showing  the  points 
on  the  diagrams  corresponding  to  each  of  the  24  crank  positions; 
these  lines  are  shown  very  light  on  the  diagrams.  Next,  draw 
on  the  diagrams  the  lines  of  zero  pressure  which  are  parallel 
to  the  atmospheric  lines  and  at  distances  below  them  equal  to  the 
atmospheric  pressure  on  scale  s. 

Having  done  this  preliminary  work,  it  is  next  necessary  to 
find  the  image  A'B'  of  b  for  each  of  the  24  crank  positions,  one 
of  the  images  being  shown  on  the  figure  For  the  crank  posi- 
tion 3  shown,  it  *will  be  observed  that  the  engine  is  taking  steam 
on  the  head  end  and  exhausting  on  the  crank  end,  since  the  pis- 
ton is  moving  to  the  left,  and  hence  at  this  instant  the  indicator 
pencils  would  be  at  M  and  N  on  the  head-  and  crank-end  dia- 


456     7JV8     9     lO^^^lZ  R'   14    15    16    17    18    19S20    21    22 

FIG.  101. — Crank  effort  and  torque  diagram. 

grams  respectively.  It  is  to  be  observed  that  when  the  crank 
reaches  position  21  the  piston  will  again  be  in  the  position  shown 
in  Fig.  100,  but,  since  at  that  instant  the  piston  is  moving  to  the 
right,  the  indicator  pencils  will  be  at  R  and  Q;  some  care  must 
be  taken  regarding  this  point. 

Now  let  hi  in.  represent  the  height  of  M  above  the  zero  line 
and  h2  in.  the  height  of  TV;  then  the  force  urging  the  piston  for- 
ward is  hi  X  s  X  AI,  while  that  opposing  it  is  hz  X  s  X  Az  and 
hence  the  piston  is  moving  forward  under  a  positive  net  force  of 

P  =  hi  X  s  X  AI  -  h2  X  s  X  A2  pds. 

While  it  is  clear  that  P  is  positive  in  this  position,  and  as  a  matter 
of  fact  is  positive  for  most  crank  positions,  yet  there  are  some  in 
which  it  is  negative,  the  meaning  of  which  is  that  in  these  posi- 
tions the  mechanism  has  to  force  the  piston  to  move  against 
an  opposing  steam  pressure;  the  mechanism  is  able  to  do  this 
to  a  limited  extent  by  means  of  the  energy  stored  up  in  its  parts. 
From  the  value  of  P  thus  found,  the  crank  effort  E  is  deter- 
mined by  the  method  already  explained  and  the  process  repeated 


CRANK-EFFORT  AND  TURNING-MOMENT       169 

for  each  of  the  24  crank  positions,  obtaining  in  this  way  24 
values  of  E.  These  will  be  found  to  vary  within  fairly  wide 
limits.  Then,  using  the  axis  of  Fig.  101,  having  a  base  OX  equal 
the  circumference  of  the  crankpin  circle,  plot  the  values  of  E 
thus  found  at  each  of  the  24  positions  marked  and  in  this  way 
the  crank-effort  diagram  OMNRSX  is  found,  vertical  heights 
on  the  diagram  representing  crank  efforts  for  the  corresponding 
crankpin  positions,  and  these  heights  may  also  be  taken  to  repre- 
sent the  torques  on  a  proper  scale  determined  from  the  crank- 
effort  scale. 

158.  Relation  between  Crank-effort  and  Indicator  Diagrams. 
—From  its  construction,  horizontal  distances  on  the  crank-effort 
diagram  represent  space  in  feet  travelled  by  the  pin,  while  verti- 
cal distances  represent  forces  in  pounds,  in  the  direction  of 
motion  of  the  crankpin,  and  therefore  the  area  under  this  curve 
represents  the  work  done  on  the  crankshaft  in  foot-pounds. 
Since  the  areas  of  the  indicator  diagrams  represent  foot-pounds 
of  work  delivered  to  the  piston,  and  from  it  to  the  crank,  there- 
fore the  work  represented  by  the  indicator  diagrams  must  be 
exactly  equal  to  that  represented  by  the  crank-effort  diagram. 
The  stroke  of  the  piston  has  been  taken  as  L  ft.  and  hence  the 
length  of  the  base  OX  will  represent  TT  X  L  ft.,  while  the  length 
of  each  indicator  diagram  will  represent  L  ft.  Calling  pm  the 
mean  pressure  corresponding  to  the  two  diagrams  and  EM  the 
mean  crank  effort,  then  2  L  X  pm=  TT  X  L  X  EM  ft.-pds.  or 

2 
EM  —  -pm  pds.,  that  is,  the  mean  height  of  the  crank-effort  diagram 

2 

in  pounds  is  -  times  the  mean  indicated  pressure  as  shown  by 

the  indicator  diagrams.  In  this  way  the  mean  crank-effort 
line  LU  may  be  located,  and  this  location  may  be  checked  by 
finding  the  area  under  the  crank-effort  diagram  in  foot-pounds, 
by  planimeter  and  then  dividing  this  by  OX  will  give  EM. 

The  crank-effort  diagram  may  also  be  taken  to  represent 
torques.  Thus,  if  the  diagram  is  drawn  on  a  vertical  scale  of  E 
pds.  equal  1  in.,  and  if  the  crank  radius  is  a  ft.  then  torques  may  be 
scaled  from  the  diagram  using  a  scale  of  E  X  a  ft.-pds.  equal  to 
1  in. 

The  investigation  above  takes  no  account  of  the  effect  of  inertia 
of  the  parts  as  this  matter  is  treated  extensively  in  Chapter  XV 
under  accelerations  in  machinery. 


170 


THE  THEORY  OF  MACHINE'S 


159.  Various  Types  of  Steam  Engines. — An  examination  of 
Fig.  101  shows  that  the  turning  moment  on  the  crankshaft,  in 
the  engine  discussed,  is  very  variable  indeed  and  this  would 
cause  certain  variations  in  the  operation  of  the  engine  which 
will  be  discussed  later.  In  the  meantime  it  may  be  stated  that 
designers  try  to  arrange  the  machinery  as  far  as  possible  to  pro- 
duce uniform  effort  and  torque. 


Resultant  Torque 
Mean    /    /~\  Torque 


FIG  102. — Torque  diagrams  for  cross-compound  engine. 

Steam  engines  are  frequently  designed  with  more  than  one 
cylinder,  sometimes  as  compound  engines  and  sometimes  as 
twin  arrangements,  as  in  the  locomotive  and  in  many  rolling-mill 
engines.  Compound  engines  may  have  two  or  three  and  some- 
times four  expansions,  requiring  at  least  two,  three  or  four  cylin- 
ders, respectively.  Engines  having  two  expansions  are  arranged 
either  with  the  cylinders  tandem  and  having  both  pistons 


FIG.  103. — Torque  diagrams  for  tandem  engine. 

connected  to  the  same  crosshead,  or  as  cross-compound  engines 
with  the  cylinders  placed  side  by  side  and  each  connected  through 
its  own  crosshead  and  connecting  rod  to  the  one  crankshaft, 
the  cranks  being  usually  of  the  same  radius  and  being  set  at  90° 
to  one  another.  In  Fig.  102  are  shown  torque  diagrams  for  twin 
engines  as  used  in  the  locomotive  or  for  a  cross-compound  engine 
with  cranks  at  90°,  the  curve  A  showing  the  torque  corresponding 


CRANK-EFFORT  AND  TURNING-MOMENT       171 

to  the  high-pressure  cylinder  with  leading  crank  and  B  that  for 
the  low-pressure  cylinder,  while  the  curve  C  in  plain  lines  gives 
the  resultant  torque  on  the  crankshaft,  and  the  horizontal  dotted 
line  D  shows  the  corresponding  mean  torque.  The  very  great 
improvement  in  the  torque  diagram  resulting  from  this  arrange- 
ment of  the  engine  is  evident,  for  the  torque  diagram  C  varies  very 
little  from  the  mean  line  D  and  is  never  negative  as  it  was  with 
the  single-cylinder  engine. 

On  the  other  hand,  the  tandem  engine  shows  no  improvement 
in  this  respect  over  the  single-cylinder  machine  as  is  shown  by 
the  torque  diagram  corresponding  to  it  shown  in  Fig.  103,  the 
dotted  curves  corresponding  to  the  separate  cylinders  and  the 
plain  curve  being  the  resultant  torque  on  the  shaft. 


FIG.  104. — Torque  diagram  for  triple-expansion  engine. 

Increasing  the  number  of  cylinders  and  cranks  usually  smooths 
out  the  torque  curve  and  Fig.  104  gives  the  results  obtained  from 
a  triple-expansion  engine  with  cranks  set  at  120°,  in  which  it  is 
seen  that  the  curve  of  mean  torque  differs  very  little  from  the 
actual  torque  produced  by  the  cylinders. 

160.  Internal-combustion  Engines. — It  will  be  well  in  con- 
nection with  this  question  to  examine  its  bearing  on  internal- 
combustion  engines,  now  so  largely  used  on  self-propelled  vehicles 
of  all  kinds.  Internal-combustion  engines  are  of  two  general 
classes,  two-cycle  and  four-cycle,  and  almost  all  machines  of  this 
class  are  single-acting,  and  only  such  machines  are  discussed 
here  as  the  treatment  of  the  double-acting  engine  offers  no 
difficulties  not  encountered  in  the  present  case. 

In  the  case  of  four-cycle  engines  the  first  outward  stroke  of 
the  piston  draws  in  the  explosive  mixture  which  is  compressed  in 
the  return  stroke.  At  the  end  of  this  stroke  the  charge  is  ignited 
and  the  pressure  rises  sufficiently  to  drive  the  piston  forward  on 


172  THE  THEORY  OF  MACHINES 

the  third  or  power  stroke,  on  the  completion  of  which  the  exhaust 
valve  opens  and  the  burnt  products  of  combustion  are  driven  out 
by  the  next  instroke  of  the  piston.  Thus,  there  is  only  one 
power  stroke  (the  third)  for  each  four  strokes  of  the  piston,  or 
for  each  two  revolutions.  An  indicator  diagram  for  this  type  of 
engine  is  shown  in  (a)  Fig.  105,  and  the  first  and  fourth  strokes 
are  represented  by  straight  lines  a  little  below  and  a  little 
above  the  atmospheric  line  respectively. 

The  indicator  diagram  from  a  two-cycle  engine  is  also  shown 
in  (6)  Fig.  105  and  differs  very  little  from  the  four-cycle  card 


(a) 


FIG.  105. — Gas-engine  diagrams. 

except  that  the  first  and  fourth  strokes  are  omitted.  The  action 
of  this  type  may  be  readily  explained.  Imagine  the  piston  at  its 
outer  end  and  the  cylinder  containing  an  explosive  mixture, 
then  as  the  piston  moves  in  the  charge  is  compressed,  ignited 
near  the  inner  dead  point,  and  this  forces  the  piston  out  on  the 
next  or  power  stroke.  Near  the  end  of  this  stroke  the  exhaust  is 
opened  and  the  burnt  gases  are  displaced  and  driven  out  by  a 
fresh  charge  of  combustible  gas  which  is  forced  in  under  slight 
pressure;  this  charge  is  then  compressed  on  the  next  instroke. 
In  this  cycle  there  is  one  power  stroke  to  each  two  strokes  of  the 
piston  or  to  each  revolution,  and  thus  the  machine  gets  the  same 
number  of  power  strokes  as  a  single-acting  steam  engine. 

The  torque  diagram  for  a  four-cycle  engine  is  shown  in  Fig. 
106  and  its  appearance  is  very  striking  as  compared  with  those 
for  the  steam  engine,  for  evidently  the  torque  is  negative  for 
three  out  of  the  four  strokes,  that  is  to  say,  there  has  to  be 
sufficient  energy  in  the  machine  parts  to  move  the  piston  during 
these  strokes,  and  all  the  energy  is  supplied  by  the  gas  through 
the  one  power  or  expansion  stroke.  The  torque  has  evidently 
very  large  variations  and  the  total  resultant  mean  torque  is  very 
small  indeed. 

For  the  two-cycle  engine  the  torque  diagram  will  be  similar  to 
the  part  of  the  curve  shown  in  Fig.  106  and  included  in  the  com- 
pression and  expansion  strokes,  the  suction  and  exhaust  strokes 


CRANK-EFFORT  AND  TURNING-MOMENT       173 

being  omitted.     Evidently  also  the  mean  torque  line  will  be 
much  higher  than  for  the  four-cycle  curve. 

Returning  now  to  the  four-cycle  engine  it  is  seen  that  the  turn- 
ing moment  is  very  irregular  and  if  such  an  engine  were  used 
with  a  small  flywheel  in  driving  a  motorcycle  or  dynamo,  the 
motion  would  be  very  unsteady  indeed,  and  would  give  so  much 
trouble  that  some  special  means  must  be  used  to  control  it. 
Various  methods  are  taken  of  doing  this,  one  of  the  most  common 


Suction  Stroke  i     Compression  Stroke      .      Expansion  Stroke         I         Exhaust  Stroke 


FIG.  106. — Torque  diagram  for  four-cycle  gas  engine. 

in  automobiles,  etc.,  being  to  increase  the  number  of  cylinders. 
Torque  diagrams  from  two  of  the  more  common  arrangements 
are  shown  in  Fig.  107.  The  diagram  marked  (a)  gives  the 
results  for  a  two-cylinder  engine  where  these  are  either  opposed 
or  are  placed  side  by  side  and  the  cranks  are  at  180°.  Diagram 
(6)  gives  the  results  from  a  four-cylinder  engine  and  corresponds 


FIG.  107. — Torque  diagrams  for  multicylinder  gasoline  engines. 

to  the  use  of  two  opposed  engines  on  the  same  shaft  or  of  four 
cylinders  side  by  side,  each  crank  being  180°  from  the  one  next  it. 
All  of  the  arrangements  shown  clearly  raise  the  line  of  mean 
torque  and  thus  make  the  irregularities  in  the  turning  moment 
very  much  less,  and  by  sufficiently  increasing  the  number  of 
cylinders  this  moment  may  be  made  very  regular.  Some  auto- 
mobiles now  have  twelve-cylinder  engines  resulting  in  very 
uniform  turning  moment  and  much  steadiness,  and  the  arrange- 


174  THE  THEORY  OF  MACHINES 

ments  made  of  cylinders  in  aeroplanes  are  particularly  satis- 
factory. Space  does  not  permit  the  discussion  of  the  matter  in 
any  further  detail  as  the  subject  is  one  which  might  profitably 
form  a  subject  for  a  special  treatise. 

161.  General  Discussion  on  Torque  Diagrams. — The  unsteadi- 
ness resulting  from  the  variable  nature  of  the  torque  has  been 
referred  to  already  and  may  now  be  discussed  more  in  detail, 
although  a  more  complete  treatment  of  the  subject  will  be  found 
in  Chapter  XIII  under  the  heading  of  "  Speed  Fluctuations  in 
Machinery." 

For  the  purpose  of  the  discussion  it  is  necessary  to  assume  the 
kind  of  load  which  the  engine  is  driving,  and  this  affects  what  is 
to  be  said.  Air  compressors  and  reciprocating  pumps  produce 
variable  resisting  torques,  the  diagram  representing  the  torque 
required  to  run  them  being  somewhat  similar  to  that  of  the  engine, 
as  shown  in  Fig.  101.  It  will  be  assumed  here,  however,  that  the 
engine  is  driving  a  dynamo  or  generator,  or  turbine  pump,  or 
automobile  or  some  machine  of  this  nature  which  requires  a 
constant  torque  to  keep  it  moving;  then  the  torque  required 
for  the  load  will  be  that  represented  by  the  mean  torque  line  in 
the  various  figures,  and  this  mean  torque  is  therefore  also  what 
might  be  called  the  load  curve  for  the  engine. 

Consider  Fig.  101 ;  it  is  clear  that  at  the  beginning  of  the  revolu- 
tion the  load  is  greater  than  the  torque  available,  whereas  be- 
tween M  and  N  the  torque  produced  by  the  engine  is  in  excess 
of  the  load  and  the  same  thing  is  true  from  R  to  S,  while  NR 
and  SU  on  the  other  hand  represent  times  when  the  load  is  in 
excess.  Further,  the  area  between  the  torque  curve  and  MN 
plus  the  corresponding  area  above  RS  represents  the  total  work 
which  the  engine  is  able  to  do,  during  these  periods,  in  excess 
of  the  load,  and  must  be  equal  to  the  sum  of  the  areas  between 
LM,  NR  and  SU  and  the  torque  curve. 

Now  during  MN  the  excess  energy  must  be  used  up  in  some 
way  and  evidently  the  only  way  is  to  store  up  energy  in  the  parts 
of  the  machine  during  this  period,  which  energy  will  be  restored 
by  the  parts  during  the  period  NR  and  so  on.  The  net  result 
is  that  the  engine  is  always  varying  in  speed,  reaching  a  maximum 
at  N  and  S  and  minimum  values  at  M  and  R}  and  the  amount  of 
these  speed  variations  will  depend  upon  the  mass  of  the  moving 
parts.  It  is  always  the  purpose  of  the  designer  to  limit  these 
variations  to  the  least  practical  amounts  and  the  torque  curves 


CRANK-EFFORT  AND  TURNING-MOMENT       175 

show  one  means  of  doing  this.  Thus,  the  tandem  compound 
engine  has  a  very  decided  disadvantage  relative  to  the  cross- 
compound  engine  in  this  respect. 

Internal-combustion  engines  with  one  cylinder  are  also  very 
deficient  because  Fig.  106  shows  that  the  mean  torque  is  only  a 
small  fraction  of  the  maximum  and  further  that  enough  energy 
must  be  stored  up  in  the  moving  parts  during  the  expansion 
stroke  to  carry  the  engine  over  the  next  three  strokes.  When 
such  engines  are  used  with  a  single  cylinder  they  are  always 
constructed  with  very  heavy  flywheels  in  order  that  the  parts 
may  be  able  to  store  up  a  large  amount  of  energy  without  too 
great  variation  in  speed.  In  automobiles  large  heavy  wheels 
are  not  possible  and  so  the  makers  of  these  machines  always  use 
a  number  of  cylinders,  and  in  this  way  stamp  out  very  largely 
the  cause  of  the  difficulty.  This  is  very  well  shown  in  the  figures 
representing  the  torques  from  multicylinder  engines,  and  in  such 
engines  it  is  well  known  that  the  action  is  very  smooth  and  even 
and  yet  all  parts  of  the  machine  including  the  flywheel  are 
quite  light.  It  has  not  yet  been  possible,  however,  to  leave  off 
the  flywheel  from  these  machines. 

QUESTIONS  ON  CHAPTER  X 

1.  Using  the  data  given  in  the  engine  of  Chapter  XIII  and  the  indicator 
diagrams  there  given,  plot  the  crank  effort  and  torque  diagrams.     For  what 
crank  angle  are  these  a  maximum? 

2.  What  would  be  the  torque  curve  for  two  engines  similar  to  that  in 
question  1,  with  cranks  coupled  at  90°?     At  what  crank  angle  would  these 
give  the  maximum  torque? 

3.  Compare  the  last  results  with  two  cranks  at  180°  and  three  cranks  at 
120°. 

4.  Using  the  diagram  Fig.  161  and  the  data  connected  therewith,  plot  the 
crank-effort  curve. 

5.  In  an  automobile  motor  3)^  in.  bore  and  5  in.  stroke,  the  rod  is  12  in. 
long;  assuming  that  the  diagram  is  similar  to  Fig.  161,  but  the  pressures  are 
only  two-thirds  as  great,  draw  the  crank-effort  curve  and  torque  diagram. 
Draw  the  resulting  curve  for  two  cylinders,  cranks  at  180°;  four  cylinders, 
cranks  at  90°;  six  cylinders,  cranks  at  60°  and  at  120°  respectively.     Try  the 
effect  of  different  sequence  of  firing. 


CHAPTER  XI 
THE  EFFICIENCY  OF  MACHINES 

162.  Input  and  Output. — The  accurate  determination  of  the 
efficiency  of  machines  and  the  loss  by  friction  is  extremely  com- 
plicated and  difficult,  and  it  is  doubtful  whether  it  is  possible 
to  deal  with  the  matter  except  through  fairly  close  approxima- 
tions. All  machines  are  constructed  for  the  purpose  of  doing 
some  specific  form  of  work,  the  machine  receiving  energy  in 
one  form  and  delivering  this  energy,  or  so  much  of  it  as  is  not 
wasted,  in  some  other  form;  thus,  the  water  turbine  receives 
energy  from  the  water  and  transforms  the  energy  thus  received 
into  electrical  energy  by  means  of  a  dynamo;  or  a  motor  receives 
energy  from  the  electric  circuit,  and  changes  this  energy  into 
that  necessary  to  drive  an  automobile,  and  so  for  any  machine. 
For  convenience,  the  energy  received  by  the  machine  will  be 
referred  to  as  the  input  and  the  energy  delivered  by  the  machine 
as  the  output. 

Now  a  machine  cannot  create  energy  of  itself,  but  is  only  used 
to  change  the  form  of  the  available  energy  into  some  other  which 
is  desired,  so  that  for  a  complete  cycle  of  the  machine  (e.g.,  one 
revolution  of  a  steam  engine,  or  two  revolutions  of  a  four-cycle 
gas  engine  or  the  forward  and  return  stroke  of  a  shaper)  there 
must  be  some  relation  between  the  input  and  the  output.  If 
no  energy  were  lost  during  the  transformation,  the  input  and 
output  would  be  equal  and  the  machine  would  be  perfect,  as  it 
would  change  the  form  of  the  energy  and  lose  none.  However, 
if  the  input  per  cycle  were  twice  the  output  then  the  machine 
would  be  imperfect,  for  there  would  be  a  loss  of  one-half  of  the 
energy  available  during  the  transformation.  The  output  can, 
of  course,  never  exceed  the  input.  It  is  then  the  province  of 
the  designer  to  make  a  machine  so  that  the  output  will  be  as 
nearly  equal  to  the  input  as  possible  and  the  more  nearly  these 
are  to  being  equal  the  more  perfect  will  the  machine  be. 

176 


THE  EFFICIENCY  OF  MACHINES  177 

163.  Efficiency. — In  dealing  with  machinery  it  is  customary 
to  use  the  term  mechanical  efficiency  or  efficiency  to  denote  the 
ratio  of  the  output  per  cycle  to  the  input,  or  the  efficiency  77  = 

output   per   cycle         ,  .  .     .  . 

- —  i— "     The   maximum   value  of  the   efficiency  is 

input   per   cycle 

'unity,  which  corresponds  to  the  perfect  machine,  and  the  mini- 
mum value  is  zero  which  means  that  the  machine  is  of  no  value 
in  transmitting  energy;  the  efficiency  of  the  ordinary  machine 
lies  between  these  two  limits,  electric  motors  having  an  efficiency 
of  0.92  or  over,  turbine  pumps  usually  not  over  0.80,  large  steam 
pumping  engines  over  0.90,  etc.,  while  in  the  case  where  the  clutch 
is  disconnected  in  an  automobile  engine  the  efficiency  of  the 
latter  is  zero,  all  the  input  being  used  up  in  friction. 

The  quantity  1  —  77  represents  the  proportion  of  the  input 
which  is  lost  in  the  bearings  of  the  machine  and  in  various  other 
ways;  thus  in  the  turbine  pump  above  mentioned,  77  =  0.80  and 
1  —  77  =  0.20,  or  20  per  cent,  of  the  energy  is  wasted  in  this 
case  in  the  bearings  and  the  friction  of  the  water  in  the  pump. 
The  amount  of  energy  lost  in  the  machine,  and  which  helps  to 
heat  up  the  bearings,  etc.,  will  depend  on  such  items  as  the 
nature  of  lubricant  used,  the  nature  of  the  metals  at  the  bear- 
ings and  other  considerations  to  be  discussed  later. 

Suppose  now  that  on  a  given  machine  there  is  at  any  instant 
a  force  P  acting  at  a  certain  point  on  one  of  the  links  which 
point  is  moving  at  velocity  v\  in  the  direction  and  sense  of  P; 
then  the  energy  put  into  the  machine  will  be  at  the  rate  of  Pvi 
ft.-pds.  per  second.  At  the  same  instant  let  there  be  a  resisting 
force  Q  acting  on  some  part  of  the  machine  and  let  the  point  of 
application  of  Q  have  a  velocity  with  resolved  part  v%  in  the  direc- 
tion of  Q  so  that  the  energy  output  is  at  the  rate  of  Qv2  ft.-pds. 
per  second.  The  force  P  may  for  example  be  the  pressure  acting 
on  an  engine  piston  or  the  difference  between  the  tensions  on  the 
tight  and  slack  sides  of  a  belt  driving  a  lathe,  while  Q  may  repre- 
sent the  resistance  offered  by  the  main  belt  on  an  engine  or  by 
the  metal  being  cut  off  in  a  lathe.  Now  from  what  has  been 

already  stated  the  efficiency  at  the  instant  is  77  =  - —     -  =  ~-^ 

input        i  .v\ 

and  if  no  losses  occurred  this  ratio  would  be  unity,  but  is  always 
less  than  unity  in  the  actual  case.  Now,  as  in  practice  Qv% 
is  always  less  than  Pvi,  choose  a  force  PQ  acting  in  the  direction 
of,  and  through  the  point  of  application  of  P  such  that  PQV\  = 
12 


178  THE  THEORY  OF  MACHINES 

Qv2,  then  clearly  P0  is  the  force  which,  if  applied  to  a  friction- 
less  machine  of  the  given  type,  would  just  balance  the  resist- 
ance Q,  and 

Po 


_  _ 

"  PVl    "     PV1    ==    P 
-p 

so  that  evidently  the  efficiency  will  be  ~  at  the  instant,  and  P0 
will  always  be  less  than  P. 

The  efficiency  may  also  be  expressed  in  a  different  form.  Thus, 
let  Qo  be  the  force  which  could  be  overcome  by  the  force  P  if 
there  were  no  fricticm  in  the  machine;  then  Pvi  =  Q0vz  and 
therefore 

=  ^r  and  QQ  is  always  greater  than  Q. 


164.  Friction.  —  Whenever  two  bodies  touch  each  other  there 
is  always  some  resistance  to  their  relative  motion,  this  resistance 
being  called  friction.  Suppose  a  pulley  to  be  suitably  mounted 
in  a  frame  attached  to  a  beam  and  that  a  rope  is  over  this  pulley, 
each  end  of  the  rope  holding  up  a  weight  w  Ib.  Now,  since  each 
of  these  weights  is  the  same  they  will  be  in  equilibrium  and  it 
would  be  expected  that  if  the  slightest  amount  were  added  to 
either  weight  the  latter  would  descend.  Such  is,  however,  not 
the  case,  and  it  is  found  by  experiment  that  one  weight  may  be 
considerably  increased  without  disturbing  the  conditions  of  rest. 

It  will  also  be  found  that  the  amount  it  is  possible  to  add  to 
one  weight  without  producing  motion  will  depend  upon  such 
quantities  as  the  amount  of  the  original  weight  w,  being  greater 
as  w  increases,  the  kind  and  amount  of  lubricant  used  in  the 
bearing  of  the  pulley,  the  stiffness  of  the  rope,  the  materials  used 
in  the  bearing  and  the  nature  of  the  mechanical  work  done  on  it, 
and  upon  very  many  other  considerations  which  the  reader  will 
readily  think  of  for  himself. 

One  more  illustration  might  be  given  of  this  point.  Suppose 
a  block  of  iron  weighing  10  Ib.  is  placed  upon  a  horizontal  table 
and  that  there  is  a  wire  attached  to  this  block  of  iron  so  that  a 
force  may  be  produced  on  it  parallel  to  .the  table.  If  now  a  tension 
is  put  on  the  wire  and  there  is  no  loss  the  block  of  iron  should 
move  even  with  the  slightest  tension,  because  no  change  is  being 
made  in  the  potential  energy  of  the  block  by  moving  it  from  place 
to  place  on  the  table,  as  no  alteration  is  taking  place  in  its  height. 


THE  EFFICIENCY  OF  MACHINES  179 

It  will  be  found,  however,  that  the  block  will  not  begin  to  move 
until  considerable  force  is  produced  in  the  wire,  the  force  possibly 
running  as  high  as  1.5  pds.  The  magnitude  of  the  force  necessary 
will,  as  before,  depend  upon  the  material  of  the  table,  the  nature 
of  the  surface  of  the  table,  the  area  of  the  face  of  the  block  of 
iron  touching  the  table,  etc. 

These  two  examples  serve  to  illustrate  a  very  important  matter 
connected  with  machinery.  Taking  the  case  of  the  pulley,  it  is 
found  that  a  very  small  additional  weight  will  not  cause  motion, 
and  since  there  must  always  be  equilibrium,  there  must  be  some 
resisting  force  coming  into  play  which  is  exactiy  equal  to  that 
produced  by  the  additional  weight.  As  the  additional  weight 
increases,  the  resisting  force  must  increase  by  the  same  amount, 
but  as  the  additional  weight  is  increased  more  and  more  the  resist- 
ing force  finally  reaches  a  maximum  amount,  after  which  it 
is  no  longer  able  to  counteract  the  additional  weight  and  then 
motion  of  the  weights  begins.  There  is  a  peculiarity  about  this 
resisting  force  then,  it  begins  at  zero  where  the  weights  are  equal 
and  increases  with  the  inequality  of  the  weights  but  finally  reaches 
a  maximum  value  for  a  certain  difference  between  them,  and  if 
the  difference  is  increased  beyond  this  amount  the  weights  move 
with  acceleration. 

In  the  case  of  the  block  of  iron  on  the  table  something  of  the 
same  nature  occurs.  At  first  there  is  no  tension  in  the  wire  and 
therefore  no  resisting  force  is  necessary,  but  as  the  tension  in- 
creases the  resisting  force  must  also  increase,  finally  reaching  a 
maximum  value,  after  which  it  is  no  longer  able  to  resist  the 
tension  produced  in  the  wire  and  the  block  moves,  and  the 
motion  of  the  block  will  be  accelerated  if  the  tension  is  still 
further  increased.  This  resisting  force  must  be  in  the  direction 
of  the  force  in  the  wire  but  opposite  in  sense,  so  that  it  must  act 
parallel  to  the  table,  that  is,  to  the  relative  direction  of  sliding, 
and  increases  from  zero  to  a  limiting  value. 

The  resisting  force  referred  to  above  always  acts  in  a  way  to 
oppose  motion  of  the  parts  and  also  always  acts  tangent  to  the 
surfaces  in  contact,  and  to  this  resisting  force  the  name  of  friction 
has  been  applied.  Much  discussion  has  taken  place  as  to  the 
nature  of  the  force,  or  whether  it  is  a  force  at  all,  but  for  the 
present  discussion  this  idea  will  be  adopted  and  this  method  of 
treatment  will  give  a  satisfactory  solution  of  all  problems  con- 
nected with  machinery. 


180  THE  THEORY  OF  MACHINES 

Wherever  motion  exists  friction  is  always  acting  in  a  sense 
opposed  to  the  motion,  although  in  many  cases  its  very  presence 
is  essential  to  motion  taking  place.  Thus  it  would  be  quite 
impossible  to  walk  were  it  not  for  the  friction  between  one's 
feet  and  the  earth,  a  train  could  not  run  were  there  no  friction 
between  the  wheels  and  rails,  and  a  belt  would  be  of  no  use  in 
transmitting  power  if  there  were  no  friction  between  the  belt 
and  pulley.  Friction,  therefore,  acts  as  a  resistance  to  motion 
and  yet  without  it  many  motions  would  be  impossible. 

165.  Laws  of  Friction. — A  great  many  experiments  have  been 
made  for  the  purpose  of  finding  the  relation  between  the  friction 
and  other  forces  acting  between  two  surfaces  in  contact.  Morin 
stated  that  the  frictional  resistance  to  the  sliding  of  one  body 
upon  another  depended  upon  the  normal  pressure  between  the 
surfaces  and  not  upon  the  areas  in  contact  nor  upon  the  velocity 
of  slipping,  and  further  that  if  F  is  the  frictional  resistance  to 
slipping  and  N  the  pressure  between  the  surfaces,  then  F  = 
nN  where  JJL  is  the  coefficient  of  friction  and  depends  upon  the 
nature  of  the  surfaces  in  contact  as  well  as  the  materials  composing 
these  surfaces. 

A  discussion  of  this  subject  would  be  too  lengthy  to  place  here 
and  the  student  is  referred  to  the  numerous  experiments  and 
discussions  in  the  current  engineering  periodicals  and  in  books 
on  mechanics,  such  as  Kennedy's  "  Mechanics  of  Machinery," 
and  Unwin's  "Machine  Design."  It  may  only  be  stated  that 
Morin's  statements  are  known  to  be  quite  untrue  in  the  case  of 
machines  where  the  pressures  are  great,  the  velocities  of  sliding 
high  and  the  methods  of  lubrication  very  variable,  and  special 
laws  must  be  formulated  in  such  cases.  In  machinery  the  nature 
of  the  rubbing  surfaces,  the  intensity  of  the  pressures,  the 
velocity  of  slipping,  methods  of  lubrication,  etc.,  vary  within 
very  wide  limits  and  it  has  been  found  quite  impossible  to  devise 
any  formula  that  would  include  all  of  the  cases  occurring,  or 
even  any  great  number  of  them,  when  conditions  are  so  variable. 
The  only  practical  method  seems  to  be  to  draw  up  formulas 
for  each  particular  class  of  machinery  and  method  of  lubrication. 
Thus,  before  it  is  possible  to  tell  what  friction  there  will  be  in 
the  main  bearing  of  a  steam  engine,  it  is  necessary  to  know  by 
experiment  what  laws  exist  for  the  friction  in  case  of  a  similar 
engine  having  similar  materials  in  the  shaft  and  bearing  and 
oiled  in  the  same  way,  and  if  the  machine  is  a  horizontal  Corliss 


THE  EFFICIENCY  OF  MACHINES  181 

engine  the  laws  would  not  be  the  same  as  with  a  vertical  high- 
speed engine;  again  the  laws  will  depend  upon  whether  the  lubri- 
cation is  forced  or  gravity  and  on  a  great  many  other  things. 
For  each  type  of  bearing  and  lubrication  there  will  be  a  law  for 
determining  the  frictional  loss  and  these  laws  must  in  each  case 
be  determined  by  careful  experiment. 

166.  Friction  Factor. — Following  the  method  of  Kennedy  and 
other  writers,  the  formula  used  in  all  cases  will  be  F  =  fN  for 
determining  the  frictional  force  F  corresponding  to  a  normal  pres- 
sure N  between  the  rubbing  surfaces,  where  /  is  called  the  fric- 
tion factor  and  differs  from  the  coefficient  of  friction  of  Morin 
in  that  it  depends  upon  a  greater  number  of  elements,  and  the 
law  for  /  must  be  known  for  each  class  of  surfaces,  method  of 
lubrication,   etc.,   from   a   series   of  experiments   performed   on 
similarly  constructed  and  operated  surfaces. 

In  dealing  with  machines  it  has  been  shown  that  they  are  made 
up  of  parts  united  usually  by  sliding  or  turning  pairs,  so  that  it 
will  be  well  at  first  to  study  the  friction  in  these  pairs  separately. 

FRICTION  IN  SLIDING  PAIRS 

167.  Friction  in   Sliding  Pairs. — Consider   a   pair   of  sliding 
elements  as  shown  in  Fig.  108  and  let  the  normal  component  of 
the  pressure  between  these  two  elements 

be  N,  and  let  R  be  the  resultant  external 

force    acting    upon    the    upper    element 

which  is  moving,  the  lower  one,  for  the 

present  being  considered  stationary.     Let 

the  force  R  act  parallel  to  the  surfaces  in 

the   sense   shown,    the  tendency  for  the 

body  is  then  to  move  to  the  right.     Now,  from  the  previous 

discussion,  there  is  a  certain  resistance  to  the  motion  of  a  the 

amount  of  which  is  fN,  where  /  is  the  friction  factor,  and  this 

force  must  in  the  very  nature  of  the  case   act  tangent  to  the 

surfaces  in  contact  (Sec.  164);  thus,  from  the  way  in  which  R  is 

chosen,  the  friction  force  F  =  fN  and  R  are  parallel. 

Now  if  R  is  small,  there  is  no  motion,  as  is  well  known,  for  the 
maximum  value  of  F  due  to  the  normal  pressure  N  is  greater 
than  R;  this  corresponds  to  a  sleigh  stalled  on  a  level  road,  the 
horses  being  unable  to  move  it.  If,  however,  R  be  increased 
steadily  it  reaches  a  point  where  it  is  equal  to  the  maximum 
value  of  F  and  then  the  body  will  begin  to  move,  and  so  long  as 


182  THE  THEORY  OF  MACHINES 

R  and  F  are  equal,  will  continue  to  move  at  uniform  speed  because 
the  force  R  is  just  balanced  by  the  resistance  to  motion;  this 
corresponds  to  the  case  where  the  sleigh  is  drawn  along  a  level 
road  at  uniform  speed  by  a  team  of  horses.  Should  R  be  still 
further  increased,  then  since  the  frictional  resistance  F  will  be 
less  than  R,  the  body  will  move  with  increasing  speed,  the  acceler- 
ation it  has  depending  upon  the  excess  of  R  over  F;  this  corre- 
sponds to  horses  drawing  a  sleigh  on  a  level  road  at  an  increasing 
speed,  and  just  here  it  may  be  pointed  out  that  the  friction 
factor  must  depend  upon  the  speed  in  some  way  because  the 
horses  soon  reach  a  speed  beyond  which  they  cannot  go. 
These  results  may  be  summarized  as  follows: 

1.  If  R  is  less  than  F,  that  is  R  <fN,  there  is  no  relative  motion. 

2.  If  R  is  equal  to  F,  that  is  R  =  fN,  the  relative  motion  of  the 
bodies  will  be  at  uniform  velocity. 

3.  If  R  is  greater  than  F,  that  is  R>fN,  there  will  be  accelerated 
motion,  relatively,  between  the  bodies.     R  is  the  resultant  external 
force  acting  on  the  body  and  is  parallel  to  the  surfaces  in  contact. 

Consider  next  the  case  shown  in  Fig.  109, 
where  the  resultant  external  force  R  acts 
at  an  angle  $  to  the  normal  to  the  surfaces 
in  contact,  and  let  it  be  assumed  that  the 
motion  of  a  relative  to  d  is  to  the  right  as 
shown  by  the  arrow.  The  bodies  are  taken 
to  be  in  equilibrium,  that  is,  the  velocity 
of  slipping  is  uniform  and  without  accelera- 
tion. Resolve  R  into  two  components 
AB  and  BC,  parallel  and  normal  respectively  to  the  surfaces  of 
contact,  then  since  BC  =  N  is  the  normal  pressure  between 
the  surfaces,  the  frictional  resistance  to  slipping  will  be  F  =  fN, 
from  Sec.  166,  where  /  is  the  friction  factor,  and  since  there  is 
equilibrium,  the  velocity  being  uniform,  the  value  of  F  must  be 
exactly  equal  and  opposite  to  AB,  these  two  forces  being  in  the 
same  direction.  Should  AB  exceed  F  =  fN  there  would  be  ac- 
celeration, and  should  it  be  less  than  fN  there  would  be  no  motion. 
Now  from  Fig.  109,  AB  =  R  sin  <£  and  also  AB  =  BC  tan  <£ 
=  N  tan  $.  Hence,  since  AB  =  fN,  there  results  the  relation 
fN  =  N  tan  0  or  /  =  tan  <£;  this  is  to  say,  in  order  that  two 
bodies  may  have  relative  motion  at  uniform  velocity,  the  re- 
sultant force  must  act  at  an  angle  <j>  to  the  normal  to  the  rubbing 
surfaces,  and  on  such  a  side  of  the  normal  as  to  have  a  resolved 


THE  EFFICIENCY  OF  MACHINES 


183 


part  in  the  direction  of  motion.     The  angle  <f>  is  fixed  by  the  fact 
that  its  tangent  is  the  friction  factor  /. 

168.  Angle  of  Friction. — The  angle  0  may  be  conveniently  called 
the  angle  of  friction  and  wherever  the  symbol  <£  occurs  in  the  rest 
of  this  chapter  it  stands  for  the  angle  of  friction  and  is  such  that 
its  tangent  is  the  friction  factor  /.     The  angle  <£  is,  of  course,  the 
limiting  inclination  of  the  resultant  to  the  normal  and  if  the  re- 
sultant act  at  any  other  angle  less  than  cf>  to  the  normal,  motion 
will  not  occur;  whereas  if  it  should  act  at  an  angle  greater  than  0 
there  will  be  accelerated  motion,  for  the  simple  reason  that  in  the 
latter  case,  the  resolved  part  of  the  resultant  parallel  to  the  sur- 
faces would  exceed  the  frictional  resistance,  and  there  would  then 
be  an  unbalanced  force  to  cause  acceleration. 

169.  Examples. — A  few  examples  should  make  the  principles 
clear,  and  in  those  first  given  all  friction  is  neglected  except  that 


FIG.  110.— Crosshead. 

in  the  sliding  pair.     The  friction  in  other  parts  will  be  considered 
later. 

1.  As  an  illustration,  take  an  engine  crosshead  moving  to  the 
right  under  the  steam  pressure  P  acting  on  the  piston,  Fig.  110. 
The  forces  acting  on  the  crosshead  are  the  steam  pressure  P,  the 
thrust  Q  due  to  the  connecting  rod  and  the  resultant  R  of  these 
two  which  also  represents  the  pressure  of  the  crosshead  on  the 
guide.  Now  from  the  principles  of  statics,  P,  Q  and  R  must  all 
intersect  at  one  point,  in  this  case  the  center  of  the  wristpin  0, 
since  P  and  Q  pass  through  0,  and  further  the  resultant  R  must 
be  inclined  at  an  angle  <£  to  the  normal  to  the  surfaces  in  contact, 
(Sec.  167);  thus  R  has  the  direction  shown.  Note  that  the  side 
of  the  normal  on  which  R  lies  must  be  so  chosen  that  R  has  a 
component  in  the  direction  of  motion.  Now  draw  AB  =  P 
the  steam  pressure,  and  draw  AC  and  BC  parallel  respectively  to 
R  and  Q,  then  BC  =  Q  the  thrust  of  the  rod  and  AC  =  R  the 
resultant  pressure  on  the  crosshead  shoe. 


184 


THE  THEORY  OF  MACHINES 


If  there  were  no  friction  in  the  sliding  pair  R  would  be  normal 
to  the  surface  and  in  the  triangle  ABD  the  angle  BAD  would 
be  90°;  BD  is  the  force  in  the  connecting  rod  and  AD  is  the 


pressure  on  the  shoe. 

-     -      Or  J 


The  efficiency  in  this  position  will  thus  be 
is   just   as   direct  to  find  PQ  the  force 
necessary  to  overcome  Q  if  there  were  no  friction  by  drawing  CE 

T)  T>  ft* 

normal  to  AB  then  PQ  =  BE  and  rj  =  -73-  =  TTJ* 

Jr          t5J\. 
2.  A  cotter  is  to  be  designed  to  connect  two  rods,  Fig.  Ill; 


FIG.   111.— Cotter  pin. 


it  is  required  to  find  the  limiting  taper  of  the  cotter  to  prevent  it 
slipping  out  when  the  rod  is  in  tension.  It  will  be  assumed  that 
both  parts  of  the  joint  have  the  same  friction  factor/,  and  hence 
the  same  friction  angle  4>,  and  that  the  cotter  tapers  only  on 
one  side  with  an  angle  6.  The  sides  of  the  cotter  on  which  the 
pressure  comes  are  marked  in  heavy  lines  and  on  the  right-hand 
side  the  total  pressure  Ri  is  divided  into  two  parts  »by  the  shape 
of  the  outer  piece  of  the  connection.  Both  the  for-ces  R\  and 
Rz  act  at  angle  <£  to  the  normal  to  their  surfaces  and,  from  what 
has  already  been  said,  it  will  be  understood  that  when  the  cotter 
just  begins  to  slide  out  they  act  on  the  side  of  the  normal  shown, 
so  that  by  drawing  the  vector  triangle  on  the  left  of  height 
AB  =  P  and  having  CB  and  BD  respectively  parallel  to  Ri 
and  R2)  the  force  Q  necessary  to  force  the  cotter  out  is  given  by 
the  side  CD. 


THE  EFFICIENCY  OF  MACHINES  185 

In    the    figure    the    angle  ABC   =   0    and    ABD  =  0   —  0. 
Therefore 

Q  =  P  [tan  0  +  tan  (0  -  6)] 
The  cotter  will  slip  out  of  itself  when  Q  =  0,  that  is 

tan  0  +  tan  (0  —  6)  =  0, 
or  0  =  20 

This  angle  0  is  evidently  independent  of  P  except  in  so  far  as  0 
is  affected  by  the  tension  P  in  the  rod. 


FIG.  112. — Lifting  jack. 

If  the  cotter  is  being  driven  in  then  the  sense  of  the  relative 
motion  of  the  parts  is  reversed  and  hence  the  forces  Ri  and  Rz 
take  the  directions  Ri  and  RJ  and  the  vector  diagram  for 
this  case  is  also  shown  on  the  right  in  the  figure.  The  force 
Q'  =  C'D'  necessary  to  drive  the  cotter  in  is  Q'  =  P[tan  0  + 
tan  (0  +  0)]  and  Q'  increases  with  6.  Small  values  of  6  make 
the  cotter  easy  to  drive  in  and  harder  to  drive  out. 

3.  An  interesting  example  of  the  friction  in  sliding  connections 
is  given  in  Fig.  112,  which  shows  a  jack  commonly  used  in  lifting 
automobiles,  etc.;  the  outlines  of  the  jack  only  are  shown,  and 


186  THE  THEORY  OF  MACHINES 

no  details  shown  of  the  arrangement  for  lowering  the  load.  In  the 
figure  the  force  P  applied  to  lifting  the  load  Q  on  the  jack  is  as- 
sumed to  act  in  the  direction  of  the  pawl  on  the  end  of  the  handle, 
and  this  would  represent  its  direction  closely  although  the  direc- 
tion of  P  will  vary  with  each  position  of  the  handle.  The  load 
Q  is  assumed  applied  to  the  toe  of  the  lifting  piece,  and  when  the 
load  is  being  raised  the  heel  of  the  moving  part  presses  against 
the  body  of  the  jack  with  a  force  R\  in  the  direction  shown  and 
the  top  pressure  between  the  parts  is  R2,  both  R i  and  J?2  being 
inclined  to  the  normals  at  angle  <£. 

At  the  base  of  the  jack  are  the  forces  Q  and  R  i,  the  resultant 
of  which  must  pass  through  A,  while  at  the  top  are  the  forces 
R  2  and  P,  the  resultant  of  which  must  pass  through  B;  and  if 
there  is  equilibrium  the  resultant  FI  of  Q  and  Ri  must  balance 
the  resultant  FI  of  R%  and  P,  which  can  only  be  the  case  if  F\ 
passes  through  A  and  B;  thus  the  direction  of  F\  is  known. 

Now  draw  the  vector  triangle  ECG  with  sides  parallel  to  F\t 
R2  and  P,  and  for  a  given  value  of  P,  so  that  F\  =  EC  and 
Rz  =  CG.  Next  through  E  draw  ED  parallel  to  Ri  and  through 
C  draw  CD  parallel  to  Q  from  which  Q  =  CD  is  found.  If  there 
were  no  friction  the  reactions  between  the  jack  and  the  frame 
would  be  normal  to  the  surfaces  at  the  points  of  contact,  thus 
A  would  move  up  to  A0  and  B  to  50  and  the  vector  diagram  would 
take  the  form  EDoCQG  where  EG  =  P  as  before  and  D0C0  =  Qo 
so  that  Qo  is  found. 

The  efficiency  of  the  device  in  this  position  is  evidently  77  =  ^r~ 

Vo- 
lt is  evident  that  with  the  load  on  the  toe,  the  efficiency  is  a 
maximum  when  the  jack  is  at  its  lowest  position  because  AB  is 
then  most  nearly  vertical,  while  for  the  very  highest  positions 
the  efficiency  will  be  low. 

4.  One  more  example  of  this  kind  will  suffice  to  illustrate  the 
principles.  Fig.  113  shows  in  a  very  elementary  form  a  quick- 
return  motion  used  on  shapers  and  machine  tools,  and  illustrated 
at  Fig.  12.  Let  Q  be  the  resistance  offered  to  the  cutting  tool 
which  is  moving  to  the  right  and  let  P  be  the  net  force  applied 
by  the  belt  to  the  circumference  of  the  belt  pulley.  For  the 
present  problem  only  the  friction  losses  in  the  sliding  elements 
will  be  considered  leaving  the  other  parts  till  later.  Here  the 
tool  holder  g  presses  on  the  upper  guide  and  the  pressure  on  this 
guide  is  Rit  the  force  in  the  rod  e  is  denoted  by  FI.  Further  the 


THE  EFFICIENCY  OF  MACHINES 


187 


pressure  of  6  on  c  is  to  the  right  and  as  the  former  is  moving 
downward  for  this  position  of  the  machine,  the  direction  of 
pressure  between  the  two  is  R2  through  the  center  of  the  pin. 

Now  on  the  driving  link  a  the  forces  acting  are  P  and  R2,  the 
resultant  F2  of  which  must  pass  through  0  and  A.  In  the  vector 
diagram  draw  BC  equal  and  parallel  to  P,  then  CD  and  BD 
parallel  respectively  to  F2  and  R2  will  represent  these  two  forces 


Q        H 
PIG.  113. — Quick-return  motion. 


\HQ 


so  that  R2  is  determined.  Again  on  c  the  forces  acting  are  R? 
and  FI,  and  their  resultant  passes  through  Oi  and  also  through  E, 
the  intersection  of  FI  and  R2,  so  that  drawing  BG  and  DG  in  the 
vector  diagram  parallel  respectively  to  F3  and  FI  gives  the  force 
FI  =  DG  in  the  rod  e.  Acting  on  the  tool  holder  g  are  the  forces 
FI,  Q  and  R i  and  the  directions  of  them  are  known  and  also  the 
magnitude  of  FI,  hence  complete  the  triangle  GHD  with  sides 
parallel  to  the  forces  concerned  and  then  GH  =  Q  and  HD  =  R! 
which  gives  at  once  the  resistance  Q  which  can  be  overcome  at 
the  tool  by  a  given  net  force  P  applied  by  the  belt. 

If  there  were  no  friction  in  these  sliding  pairs  then  the  forces 
RI  and  R%  would  act  normal  to  the  sliding  surfaces  instead  of 
at  angles  fa  and  <f>2  to  the  normals  so  that  A  moves  to  AQ  and 


188 


THE  THEORY  OF  MACHINES 


E  to  EQ  and  the  construction  is  shown  by  the  dotted  lines,  from 
which  the  value  of  Q0  is  obtained.     The  efficiency  for  this  posi- 


The  value  of  rj  should  be  found 


tiori  of  the  machine  is  17  =  -~r 

Vo 

for  a  number  of  other  positions  of  the  machine,  and,  if  desirable, 
a  curve  may  be  plotted  so  that  the  effect  of  friction  may  be 
properly  studied. 

Before  passing  on  to  the  case  of  turning  pairs  the  attention  of 
the  reader  is  called  to  the  fact  that  the  greater  part  of  the  problem 
is  the  determination  of  the  condition  of  static  equilibrium  as 
described  in  Chapter  IX,  the  method  of  solution  being  by  means 
of  the  virtual  center,  in  these  cases  the  permanent  center  being 
used.  The  only  difficulty  here  is  in  the  determination  of  the 

direction  of  the  pressures  R 
between  the  sliding  surfaces, 
and  the  following  suggestions 
may  be  found  helpful  in  this 
regard. 

Let  a  crosshead  a,  Fig.  114, 
slide  between  the  two  guides 
di  and  d2,  first  find  out,  by 
inspection  generally,  from  the 
forces  acting  whether  the 
pressure  is  on  the  guide  di 
or  d2.  Thus  if  the  con- 
necting rod  and  piston  rod  are  in  compression  the  pressure 
is  on  d2,  if  both  are  in  tension  it  is  on  di,  etc.,  suppose  for  this 
case  that  both  are  in  compression,  the  heavy  line  showing  the 
surface  bearing  the  pressure. 

Next  find  the  relative  direction  of  sliding.  It  does  not  matter 
whether  both  surfaces  are  moving  or  not,  only  the  relative 
direction  is  required  it  is  assumed  in  the  sense  shown,  i.e.,  the 
sense  of  motion  of  a  relative  to  cZ2  is  to  the  left  (and,  of  course,  the 
sense  of  motion  of  d2  relative  to  a  is  to  the  right).  Now  the  re- 
sultant pressure  between  the  surfaces  is  inclined  at  angle  <f> 
to  the  normal  where  $  =  tan"1/,  /  being  the  friction  factor,  so 
that  the  resultant  must  be  either  in  the  direction  of  R  i  or  R\. 

Now  RI  the  pressure  of  a  upon  d!2  acts  downward,  and  in 
order  that  it  may  have  a  resolved  part  in  the  direction  of  motion, 
then  RI  and  not  RI  is  the  correct  direction.  If  RI  is  treated  as 
the  pressure  of  dz  upon  a  then  RI  acts  upward,  but  the  sense  of 


THE  EFFICIENCY  OF  MACHINES 


189 


motion  of  d2  relative  to  a  is  the  opposite  of  that  of  a  relative  to 
dz,  and  hence  from  this  point  of  view  also  Ri  is  correct. 

It  is  easy  to  find  the  direction  of  RI  by  the  following  simple 
rule:  Imagine  either  of  the  sliding  pieces  to  be  an  ordinary 
carpenter's  wood  plane,  the  other  sliding  piece  being  the  wood 
to  be  dressed,  then  the  force  will  have  the  same  direction  as  the 
tongue  of  the  plane  when  the  latter  is  being  pushed  in  the  given 
direction  on  the  cutting  stroke,  the  angle  to  the  normal  to  the 
surfaces  being  </>. 

170.  Turning  Paks. — In  dealing  with  turning  pairs  the  same 
principles  are  adopted  as  are  used  with  the  sliding  pairs  and  should 
not  cause  any  difficulty.  Let  a,  Fig.  115,  represent  the  outer 


FIG.  115. 

element  of  a  turning  pair,  such  as  a  loose  pulley  turning  in  the 
sense  shown  upon  the  fixed  shaft  d  of  radius  r,  and  let  the  forces 
P  and  Q  act  upon  the  outer  element.  It  must  be  explained  that 
the  arrow  shows  the  sense  in  which  the  pulley  turns  relatively 
to  the  shaft  and  this  is  to  be  understood  as  the  meaning  of  the 
arrow  in  the  rest  of  the  present  discussion.  It  may  be  that  both 
elements  are  turning  in  a  given  case,  and  the  two  elements  may 
also  turn  in  the  same  or  in  opposite  sense,  but  the  arrow  indicates 
the  relative  sense  of  motion  and  the  forces  P  and  Q  are  assumed 
to  act  upon  the  link  on  which  they  are  drawn,  that  is  upon  a 
in  Fig.  115. 


190 


THE  THEORY  OF  MACHINES 


If  there  were  no  friction  then  the  resultant  of  P  and  Q  would 
pass  through  the  intersection  A  of  these  forces  and  also  through 
the  center  0  of  the  bearing,  so  that  under  these  circumstances  it 
would  be  possible  to  find  Q  for  a  given  value  of  P  by  drawing  the 
vector  triangle. 

There  is,  however,  frictional  resistance  offered  to  motion  at  the 
surface  of  contact,  hence  if  the  resultant  R  of  P  and  Q  acted 
through  0,  there  could  be  no  motion.  In  order  that  motion  may 
exist  it  is  necessary  that  the  resultant  produce  a  turning  moment 
about  the  center  of  the  bearing  equal  and  opposite  to  the  resist- 
ance offered  by  the  friction  between  the  surfaces.  It  is  known 

already  that  the  frictional 
resistance  is  of  such  a  na- 
ture as  to  oppose  motion, 
and  hence  the  resultant  force 
must  act  in  such  a  way  as 
to  produce  a  turning  mo- 
ment in  the  sense  of  mo- 
tion equal  to  the  moment 
offered  by  friction  in  the 
opposite  sense.  Thus  in 
the  case  shown  in  the 
figure  the  resultant  must 
pass  through  A  and  lie  to 
the  left  of  0. 

In  Fig.  116,  which  shows  an  enlarged  view  of  the  bearing,  let 
p  be  the  perpendicular  distance  from  0  to  R,  so  that  the  moment 
of  R  about  0  is  R  X  p.  The  point  C  may  be  conveniently 
called  the  center  of  pressure,  being  the  point  of  intersection  of  R 
and  the  surfaces  under  pressure.  Join  CO.  Now  resolve  R  into 
two  components,  the  first,  Ft  tangent  to  the  surfaces  at  C,  and 
the  second,  N,  normal  to  the  surfaces  at  the  same  point.  Fol- 
lowing the  method  employed  with  sliding  pairs,  N  is  the  normal 
pressure  between  the  surfaces  and  the  frictional  resistance  to 
.motion  will  be  fN,  where  /is  the  friction  factor  (Sec.  166),  and 
since  the  parts  are  assumed  to  be  in  equilibrium,  there  must  be 
no  unbalanced  force,  so  that  the  resolved  part  F  of  the  resultant 
R  must  be  equal  in  magnitude  to  the  frictional  resistance,  or 
F  —  fN.  But  /  =  tan  <£,  where  <j>  is  the  friction  angle,  so  that 


FIG.  116. 


tan  <f>  =  f  =  -.: 


THE  EFFICIENCY  OF  MACHINES 


191 


from  which  it  follows  that  the  angle  between  N  and  R  must 
be  <£,  and  hence  the  resultant  R  must  make  an  angle  <f>  with 
the  radius  r  at  the  center  of  pressure  C. 

171.  Friction  Circle. — With  center  0  draw  a  circle  tangent  to 
.R  as  shown  dotted ;  then  this  circle  is  the  one  to  which  the  result- 
ant R  must  be  tangent  to  maintain  uniform  relative  motion, 
and  the  circle  may  be  designated  as  the  friction  circle.  The 
radius  p  of  the  friction  circle  is  p  =  r  sin  4>,  where  r  is  the  radius 
of  the  journal,  and  this  circle  is  concentric  with  the  journal  and 
much  smaller  than  the  latter,  since  <£  is  always  a  small  angle 
in  practice.  Thus,  in  turning  pairs  the  resultant  must  always 


FIG.  117. 

be  at  an  angle  (j>  to  the  normal  to  the  surfaces,  and  this  is  most 
easily  accomplished  by  drawing  the  resultant  tangent  to  the 
friction  circle,  and  on  such  a  side  of  it  that  it  produces  a  turning 
moment  in  the  sense  of  the  relative  motion  of  the  parts.  Since 
/  is  always  small  in  actual  bearings,  <£  is  also  small,  and  hence 
tan  4>  =  sin  <£  nearly,  so  that  approximately  p  =  r  sin  <f>  =  r  tan  <£ 

=  Tf. 

Four  different  arrangements  of  the  forces  on  a  turning  pair 
are  shown  at  Fig.  117,  similar  letters  being  used  to  Fig.  115. 
At  (a)  P  and  Q  act  on  the  outer  element  but  their  resultant  R 
acts  in  opposite  sense  to  the  former  case  and  hence  on  opposite 
side  of  the  friction  circle,  since  the  relative  sense  of  rotation  is 
the  same.  In  case  (c)  P  and  Q  act  on  the  inner  element  and  the 
relative  sense  of  rotation  is  reversed  from  (a),  hence  R  passes  on 
the  right  of  the  friction  circle;  at  (fr)  conditions  are  the  same  as 
(a)  except  for  the  relative  sense  of  motion  which  also  changes  the 
position  of  R'}  at  (d)  the  forces  act  on  the  outer  element  and  the 
sense  of  rotation  and  position  of  R  are  both  as  indicated. 

172.  Examples. — The  construction  already  shown  will  be 
applied  in  a  few  practical  cases. 


192 


THE  THEORY  OF  MACHINES 


1.  The  first  case  considered  will  be  an  ordinary  bell-crank 
lever,  Fig.  118,  on  which  the  force  P  is  assumed  to  act  horizontally 
and  Q  vertically  on  the  links  a  and  c  respectively,  the  whole 
lever  b  turning  in  the  clockwise  sense.  An  examination  of  the 
figure  shows  that  the  sense  of  motion  of  a  relative  to  b  is  counter- 
clockwise as  is  also  the  motion  of  c  relative  to  6,  therefore  P 
will  be  tangent  to  the  lower  side  of  the  friction  circle  at  bearing 
1,  and  Q  will  be  tangent  to  the  left-hand  side  of  the  friction  circle 


FIG.  118. 

at  bearing  3,  and  the  resultant  of  P  and  Q  must  pass  through  A 
and  must  be  tangent  to  the  upper  side  of  the  friction  circle  on 
the  pair  2,  so  that  the  direction  of  R  becomes  fixed.  Now  draw 
DE  in  the  direction  of  P  to  represent  this  force  and  then  draw 
EF  and  DF  parallel  respectively  to  Q  and  R  and  intersecting  at 
F,  then  EF  =  Q  and  DF  =  R. 

In  case  there  was  no  friction  and  assuming  the  directions  of 
P  and  Q  to  remain  unchanged  (this  would  be  unusual  in  practice), 
then  P,  Q  and  their  resultant,  would  act  through  the  centers  of 
the  joints  1,  3  and  2  respectively.  Assuming  the  magnitude  of 
P  to  be  unchanged,  then  the  vector  triangle  DEF'  has  its  sides 
EF'  and  DF'  parallel  respectively  to  the  resistance  Q0  and  the 
resultant  RQ  so  that  there  is  at  once  obtained  the  force  QQ  =  EF'. 

Then  the  efficiency  of  the  lever  in  this  position  is  rj  =  ^-  and 

Vo 

for  any  other  position  may  be  similarly  found. 


THE  EFFICIENCY  OF  MACHINES 


193 


The  friction  circles  are  not  drawn  to  scale  but  are  made  larger 
than  they  should  be  in  order  to  make  the  drawing  clear. 

2.  Let  it  be  required  to  find  the  line  of  action  of  the  force  in 
the  connecting  rod  of  a  steam  engine  taking  into  account  friction 
at  the  crank-  and  wristpins.  To  avoid  confusion  the  details  of 
the  rod  are  omitted  and  it  is  represented  by  a  line,  the  friction 
circles  being  to  a  very  much  exaggerated  scale.  Let  Fig.  119(a) 
represent  the  rod  in  the  position  under  consideration,  the  direc- 


FIG.  119. — Steam-engine  mechanism. 

tion  of  the  crank  is  also  shown  and  the  piston  rod  is  assumed  to 
be  in  compression,  this  being  the  usual  condition  for  this  position 
of  the  crank.  Inspection  of  the  figure  shows  that  the  angle  a. 
is  increasing  and  the  angle  /?  is  decreasing,  so  that  the  line  of 
action  of  the  force  in  the  connecting  rod  must  be  tangent  to  the 
top  of  the  friction  circle  at  2  and  also  to  the  bottom  of  the'  fric- 
tion circle  at  1,  hence  it  takes  the  position  shown  in  the  light  line 
and  crosses  the  line  of  the  rod.  This  position  of  the  line  of  action 
of  the  force  is  seen  on  examination  to  be  correct,  because  in 
both  cases  the  force  acts  on  such  a  side  of  the  center  of  the  bear- 
ing as  to  produce  a  turning  moment  in  the  direction  of  relative 
motion. 

13 


194 


THE  THEORY  OF  MACHINES 


Two  other  positions  of  the  engine  are  shown  in  Fig.  119  at 
(6)  and  .(c),  the  direction  of  revolution  being  the  same  as  before 
and  the  line  of  action  of  the  force  in  the  rod  is  in  light  lines.  In 
the  case  (6),  the  rod  is  assumed  in  compression  and  evidently 
both  the  angles  a  and  /3  are  decreasing  so  that  the  line  of  action 
of  the  force  lies  below  the  axis  of  the  rod;  while  in  the  position 
shown  in  (c),  the  connecting  rod  is  assumed  in  tension,  a  is  decreas- 
ing, and  ]8  is  increasing  so  that  the  line  of  the  force  intersects  the 
rod.  In  all  cases  the  determining  factor  is  that  the  force  must 
lie  on  such  a  side  of  the  center  of  the  pin  as- to  produce  a  turning 
moment  in  the  direction  of  relative  motion. 

173.  Governor — Turning  Pairs  Only. — A  complete  device  in 
which  turning  pairs  alone  occur  is  shown  at  Fig.  120,  which 


FIG.  120.— Governor. 

represents  one  of  the  governors  discussed  fully  in  the  following 
chapter,  except  for  the  effect  of  friction.  The  governor  herewith 
is  shown  also  at  Fig.  125  and  only  one-half  of  it  has  been  drawn 
in,  the  total  weight  of  the  two  rotating  balls  is  w  Ib.  while  that 
of  the  central  weight  including  the  pull  of  the  valve  gear  is  taken 
as  W  Ib.  In  Chapter  XII  no  account  has  been  taken  of  friction 
or  pin  pressures  while  these  are  essential  to  the  present  purpose. 
There  will  be  no  frictional  resistance  between  the  central  weight 
and  the  spindle  and  the  friction  circles  at  A,  B  and  C  are  drawn 
exaggerated  in  order  to  make  the  construction  more  clear, 

It  is  assumed  that  the  balls  are  moving  slowly  outward  and 
that  when  passing  through  the  position  illustrated  the  spindle 
rotates  at  n  revolutions  per  minute  or  at  co  radians  per  second; 
it  is  required  to  find  n  and  also  the  speed  n'  of  the  spindle  as  the 


THE  EFFICIENCY  OF  MACHINES  195 

balls  pass  through  this  same  position  when  travelling  inward. 
The  difference  between  these  two  speeds  indicates  to  some  extent 
the  quality  of  the  governor,  as  it  shows  what  change  must  be 
made  before  the  balls  will  reverse  their  motion. 

On  one  ball  there  is  a  centrifugal  force  -~  pds.,  where  —  =  —  rco2, 

z  z       Zg 

C 
r  being  the  radius  of  rotation  of  the  balls  in  feet,  also  -~  acts 

w 
horizontally  while  the  weight  of  the  ball  -~  lb.   acts  vertically, 

and  their  resultant  is  a  force  P  inclined  as  shown  in  the  left-hand 
figure.  The  arms  AB  and  BC  are  both  in  tension  evidently, 
and  as  the, balls  are  moving  outward,  a.  is  increasing  and  /3  is 
decreasing  (see  Fig.  120) ;  hence  the  direction  of  the  force  in  the 
arm  BC  crosses  the  axis  of  the  latter  as  shown,  F\  representing 
the  force. 

Now  the  direction  of  the  force  P  is  unknown  and  it  cannot 

(j 

be  determined  without  first  finding  -=  which,  however,  depends 

z 

upon  n,  the  quantity  sought.  An  approximation  to  the  slope 
of  P  may  be  found  by  neglecting  friction  and  with  this  approxi- 
mate value  the  first  trial  may  be  made.  With  the  assumed 
direction  of  P  the  point  H,  where  P  intersects  FI,  is  determined 
and  then  the  resultant  R  of  FI  and  P  must  pass  through  H  and 
also  be  tangent  to  the  friction  circle  at  A.  (If  there  were  no 
friction,  R  would  pass  through  the  center  of  A,  Sec.  170.)  Turn- 

W 

ing  now  to  the  vector  diagram  on  the  right  make  DE  =  -^  and 

EL  =  -^r]  then  draw  DG  horizontally  to  meet  EG,  which  is  paral- 

Z 

lei  to  FI  in  G.  The  length  EG  represents  the  force  FI  in  the  arm 
BCy  while  DG  represents  the  tension  on  the  weight  W  which  is 
balanced  by  the  other  half  of  the  governor. 

Next  draw  GJ  and  EJ  parallel  respectively  to  R  and  P,  whence 
these  forces  are  found.  If  the  slope  of  P  has  been  properly 
assumed,  the  point  J  will  be  on  the  horizontal  line  through  L, 
and  if  J  does  not  lie  on  this  line  a  second  trial  slope  of  P  must  be 
made  and  the  process  continued  until  /  does  fall  on  the  horizon- 
tal through  L. 

The  length  LJ  then  represents  ~  =  ^r  ra)2  from  which  w  is 

readily  computed,  and  from  it  the  speed  n  in  revolutions  per 
minute. 


196 


THE  THEORY  OF  MACHINES 


The  dotted  lines  show  the  case  where  the  mechanism  passes 
through  the  same  position  but  with  the  balls  moving  inward 
and  from  the  length  LJ'  the  value  of  co'  and  of  n'  may  be  found. 

If  only  the  relation  between  n  and  nr  is  required,  then  —  =  A/FT/ 

n        \  LJ 

The  meaning  of  this  is  that  if  the  balls  were  moving  outward  due 
to  a  decreased  load  on  the  prime  mover  to  which  the  governor 
was  connected  then  they  would  pass  through  the  position  shown 
when  the  spindle  turned  at  n  revolutions  per  minute,  but  if  the 
load  were  again  increased  causing  the  balls  to  move  inward  the 
speed  of  the  spindle  would  have  to  fall  to  n'  before  the  balls 
would  pass  through  the  position  shown.  Evidently  the  best, 
governor  is  one  in  which  n  and  n'  most  nearly  agree,  and  the 
device  would  be  of  little  value  where  they  differed  much. 


FIG.  121. 

In  reading  this  problem  reference  should  also  be  made  to  the 
chapter  on  governors. 

174.  Machine  with  Turning  and  Sliding  Pairs. — This  chapter 
may  be  very  well  concluded  by  giving  an  example  where  both 
turning  and  sliding  pairs  are  used,  although  there  should  be  no 
difficulty  in  combining  the  principles  already  laid  down  in  any 
machine.  The  machine  considered  is  the  steam  engine,  the  barest 
outlines  of  which  are  shown  in  Fig.  121.  The  piston  is  assumed 
combined  with  the  crosshead  and  only  the  latter  is  shown,  and 
in  the  problem  it  has  been  assumed  that  the  engine  is  lifting  a 
weight  from  a  pit  by  means  of  a  vertical  rope  on  a  drum,  the 
resistance  of  the  weight  being  Q  Ib.  Friction  of  the  rope  is  not 
considered.  The  indicator  diagram  gives  the  information 
necessary  for  finding  the  pressure  P  acting  through  the  piston 


THE  EFFICIENCY  OF  MACHINES  197 

on  the  crosshead,  and  the  problem  is  to  find  Q  and  the 
efficiency. 

From  the  principles  already  laid  down,  the  direction  of  Ri 
the  pressure  on  the  crosshead  is  known,  also  the  line  of  action 
of  FI  and  of  R2.  For  equilibrium  the  forces  Fi,  R,  and  P  must 
intersect  at  one  point  which  is  evidently  A,  as  P,  the  force  due 
to  the  steam  pressure,  is  taken  to  act  along  the  center  of  the 
piston  rod.  On  the  crankshaft  there  is  the  force  F\  from  the 
connecting  rod,  and  the  force  Q  due  to  the  weight  lifted,  and  if 
there  were  no  friction,  their  resultant  would  pass  through  their 
point  of  intersection  B  and  also  through  0  the  center  of  the  crank- 
shaft. To  allow  for  friction,  however,  R2  must  be  tangent  to 
the  friction  circle  at  the  crankshaft  and  must  touch  the  top  of 
the  latter,  hence  the  position  of  R2  is  fixed.  Thus  the  locations 
of  the  five  forces,  P,  FI,  Ri,  R2  and  Q  are  known. 

Now  draw  the  vector  diagram,  laying  off  CD  *=  P  and  drawing 
CE  and  DE  parallel  respectively  to  Ri  and  FI,  which  gives  these 
two  forces,  next  draw  EF  parallel  to  R2  and  DF  parallel  to  Q 
which  thus  determines  the  magnitude  of  Q. 

If  there  were  no  friction,  FI  would  be  along  the  axis  of  the  rod, 
and  R i  normal  to  the  guides,  both  forces  passing  through  AQ  the 
center  of  the  wristpin.  Further,  R2  would  pass  through  BQ  the 
intersection  of  FI  and  Q,  it  would  also  pass  through  0  as  shown 
dotted,  so  that  the  lines  of  action  of  all  of  the  forces  are  known  and 
the  vector  diagram  CEoFoD  may  be  drawn  obtaining  the  resistance 
Qo  =  DFo,  which  could  be  overcome  by  the  pressure  P  on  the  piston 
if  there  were  no  friction.  The  efficiency  of  the  machine  in  this 

position  is  then  r>  =  -*-,  and  may  be  found  in  a  similar  way  for 
Wo 

other  positions. 

If  desired,  the  value  of  the  efficiency  for  a  number  of  positions 
of  the  machine  may  be  found  and  a  curve  plotted  similar  to  a 
velocity  diagram,  Chapter  III,  from  which  the  efficiency  per 
cycle  is  obtained. 

In  all  illustrations  the  factor  /  is  much  exaggerated  to  make 
the  constructions  clear  and  in  many  actual  cases  the  efficiency 
will  be  much  higher  than  the  cuts  show.  Where  the  efficiency 
is  very  close  to  unity,  the  method  is  not  as  reliable  as  for  low 
efficiencies,  but  many  of  the  machines  have  such  high  efficiency 
that  such  a  construction  as  described  herein  is  not  necessary, 
nor  is  any  substitute  for  it  needed  in  such  cases. 


198  THE  THEORY  OF  MACHINES 

QUESTIONS  ON  CHAPTER  XI 

1.  In  the  engine  crosshead,  Fig.  110,  if  the  friction  factor  is  0.05,  what  size 
is  the  friction  angle?     If  the  piston  pressure  is  5,000  pds.,  and  the  connect- 
ing rod  is  at  12°  to  the  horizontal,  what  is  the  pressure  in  the  rod  and  the 
efficiency  of  the  crosshead,  neglecting  friction  at  the  wristpin? 

2.  Of  two  12-in.  journals  one  has  a  friction  factor  0.002  and  the  other  0.003. 
What  are  the  sizes  of  the  friction  circles? 

3.  What  would  be  the  efficiency  of  the  crank  in  Fig.  118  if  the  scale  of  the 
drawing  is  one-quarter  and  the  pins  are  1^  in.  diameter? 

4.  Determine  the  direction  of  the  force  in  the  side  rod  of  a  locomotive  in 
various  positions. 

6.  A  thrust  bearing  like  Fig.  1  (6)  has  five  collars,  the  mean  bearing  diam- 
eter of  which  is  10  in.  If  the  shaft  runs  at  120  revolutions  per  minute  and 
has  a  bearing  pressure  of  50  Ib.  per  square  inch  of  area,  find  the  power  lost 
if  the  friction  factor  is  0.05. 

6.  In  the  engine  of  Fig.  121,  taking  the  scale  of  the  drawing  as  one-six- 
teenth and  the  friction  factor  as  0.06,  find  the  value  of  Q  when  P  =  2,500 
pds.  the  diameters  of  the  crank  and  wristpins  being  3}^  and  3  in.  respec- 
tively. 

7.  In  a  Scotch  yoke,  Fig.  6,  the  crank  is  6  in.  long  and  the  pin  2  in.  diam- 
eter, the  slot  being  3  in.  wide.     With  a  piston  pressure  of  500  pds.,  find 
the  efficiency  for  each  45°  crank  angle,  taking/  =  0.1. 


PART  II 
MECHANICS  OF  MACHINERY 


CHAPTER  XII 
GOVERNORS 

175.  Methods  of  Governing. — In  all  prime  movers,  which  will 
be  briefly  called  engines,  there  must  be  a  continual  balance  be- 
tween the  energy  supplied  to  the  engine  by  the  working  fluid  and 
the  energy  delivered  by  the  machine  to  some  other  which  it  is 
driving,  e.g.,  a  dynamo,  lathe,  etc.,  allowance  being  made  for  the 
friction  of  the  prime  mover.  Thus,  if  the  energy  delivered  by  the 
working  fluid  (steam,  water  or  gas)  in  a  given  time  exceeds  the 
sum  of  the  energies  delivered  to  the  dynamo  and  the  friction  of 
the  engine,  then  there  will  be  some  energy  left  to  accelerate  the 
latter,  and  it  will  go  on  increasing  in  speed,  the  friction  also  in- 
creasing till  a  balance  is  reached  or  the  machine  is  destroyed. 
The  opposite  result  happens  if  the  energy  coming  in  is  insufficient, 
the  result  being  that  the  machine  will  decrease  in  speed  and 
may  eventually  stop. 

In  all  cases  in  actual  practice,  the  output  of  an  engine  is  con- 
tinually varying,  because  if  a  dynamo  is  being  driven  by  it  for 
lighting  purposes  the  number  of  lights  in  use  varies  from  time  to 
time;  the  same  is  true  if  the  engine  drives  a  lathe  or  drill,  the 
demands  of  these  continually  changing. 

The  output  thus  varying  very  frequently,  the  energy  put  in 
by  the  working  fluid  must  be  varied  in  the  same  way  if  the  desired 
balance  is  to  be  maintained,  and  hence  if  the  prime  mover  is  to 
run  at  constant  speed  some  means  of  controlling  the  energy  ad- 
mitted to  it  during  a  given  time  must  be  provided. 

Various  methods  are  employed,  such  as  adjusting  the  weight  of 
fluid  admitted,  adjusting  the  energy  admitted  per  pound  of  fluid, 
or  doing  both  of  these  at  one  time,  and  this  adjustment  may  be 
made  by  hand  as  in  the  locomotive  or  automobile,  or  it  may  be 
automatic  as  in  the  case  of  the  stationary  engine  or  the  water 
turbine  where  the  adjustment  is  made  by  a  contrivance  called  a 
governor. 

A  governor  may  thus  be  defined  as  a  device  used  in  connection 
with  prime  movers  for  so  adjusting  the  energy  admitted  with  the 

201 


202  THE  THEORY  OF  MACHINES 

working  fluid  that  the  speed  of  the  prime  mover  will  be  constant 
under  all  conditions.  The  complete  governor  contains  essentially 
two  parts,  the  first  part  consisting  of  certain  masses  which  rotate 
at  a  speed  proportional  to  that  of  the  prime  mover,  and  the 
second  part  is  a  valve  or  similar  device  controlled  by  the  part 
already  described  and  operating  directly  on  the  working  fluid. 

It  is  not  the  intention  in  the  present  chapter  to  discuss  the 
valve  or  its  mechanism,  because  the  form  of  this  is  so  varied  as  to 
demand  a  complete  work  on  it  alone,  and  further  because  its 
design  depends  to  some  extent  on  the  principles  of  thermo- 
dynamics and  hydraulics  with  which  this  book  does  not  deal. 
This  valve  always  works  in  such  a  way  as  to  control  the  amount 
of  energy  entering  the  engine  in  a  given  time  and  this  is  usually 
done  in  one  of  the  following  ways: 

(a)  By  shutting  off  a  part  of  the  working  fluid  so  as  to  admit 
a  smaller  weight  of  it  per  second.  This  method  is  used  in  many 
water  wheels  and  gas  engines  and  is  the  method  adopted  in  the 
steam  engine  where  the  length  of  cutoff  is  varied  as  in  high-speed 
engines. 

(6)  By  not  only  altering  the  weight  of  fluid  admitted,  but  by 
changing  at  the  same  time  the  amount  of  energy  contained  in 
each  pound.  This  method  is  used  in  throttling  engines  of  various 
kinds. 

(c)  By  employing  combinations  of  the  above  methods  in 
various  ways,  sometimes  making  the  method  (a)  the  most  im- 
portant, sometimes  the  method  (6)  The  combined  methods 
are  frequently  used  in  gas  engines  and  water  turbines. 

The  other  part  of  the  governor,  that  is  the  one  containing  the 
revolving  masses  driven  at  a  speed  proportional  to  that  of  the 
prime  mover,  will  be  dealt  with  in  detail  because  of  the  nature 
of  the  problems  it  involves,  and  it  will  in  future  be  briefly  referred 
to  as  the  governor. 

176.  Types  of  Governors. — Governors  are  of  two  general 
classes  depending  on  the  method  of  attaching  them  to  the  prime 
mover  and  also  upon  the  disposition  of  the  revolving  masses, 
and  the  speed  at  which  these  masses  revolve.  The  first  type  of 
governors,  which  is  also  the  original  type  used  by  Watt  on  his 
engines,  has  been  named  the  rotating-pendulum  governor  because 
the  revolving  masses  are  secured  to  the  end  of  arms  pivoted  to  the 
rotating  axis  somewhat  similar  to  the  method  of  construction  of 
a  clock  pendulum,  except  that  the  clock  pendulum  swings  in  one 


GOVERNORS 


203 


Pipe 


plane,  while  the  governor  masses  revolve.  In  this  type  there  are 
three  subdivisions :  (a)  gravity  weighted,  in  which  the  centrifugal 
force  due  to  the  revolving  masses  or  balls  is  largely  balanced  by 
gravity;  (6)  spring  weighted,  in  which  the  same  force  is  largely 
balanced  by  springs;  and  (c)  combination  governors  in  which 
both  methods  are  used.  Governors  of  this  general  class  are 
usually  mounted  on  a  separate  frame  and  driven  by  belt  or  gears 
from  the  engine,  but  they  are,  at  times,  made  on  a  part  of  the 
main  shaft. 

The  second  type  is  the  inertia  governor  which  is  usually  made 
on  the  engine  power  shaft,  although  it  is  occasionally  mounted 
separately.  The  name  is  now  principally  used  to  designate  a 
class  of  governor  with  its  re- 
volving masses  differently  dis- 
tributed to  the  former  class; 
its  equilibrium  depends  on 
centrifugal  force  but  during 
the  changes  in  position  the 
inertia  of  the  masses  plays  a 
prominent  part  in  producing  Steam 
rapid  adjustment.  The  name 
shaft-governor  is  also  much 
used  for  this  type. 

177.  Revolving-pendulum 
Governor. — Beginning  with 
the  revolving-pendulum  type, 
an  illustration  of  which  is 
shown  at  Fig.  122  connected 
up  to  a  steam  engine,  it  is  seen  that  it  consists  essentially  of 
a  spindle  A,  caused  to  revolve  by  means  of  two  bevel  gears 
B  and  C,  the  latter  being  driven  in  turn  through  a  pulley  D 
which  is  connected  by  a  belt  to  the  crankshaft  of  the  engine; 
thus  the  spindle  A  will  revolve  at  a  speed  proportional  to  that 
of  the  crankshaft  of  the  engine.  To  this  spindle  at  F  two  balls 
G  are  attached  through  the  ball  arms  E,  and  these  arms  are 
connected  by  links  J  to  the  sleeve  H,  fastened  to  the  rod  R, 
which  rod  is  free  to  move  up  and  down  inside  the  spindle  A  as 
directed  by  the  movement  of  the  balls  and  links.  The  sleeve 
H  with  its  rod  R  is  connected  in  some  manner  with  the  valve  V, 
in  this  illustration  a  very  direct  connection  being  indicated,  so 
that  a  movement  of  the  sleeve  will  open  or  close  the  valve  V. 


FIG.   122. — Simple  governor. 


204 


THE  THEORY  OF  MACHINES 


The  method  of  operation  is  almost  self-evident;  as  the  engine 
increases  in  speed  the  spindle  A  also  increases  proportionately 
and  therefore  there  is  an  increased  centrifugal  force  acting  on 
the  balls  G  causing  them  to  move  outward.  As  the  balls  move 
outward  the  sleeve  H  falls  and  closes  the  valve  V  so  as  to  prevent 
as  much  steam  from  getting  in  and  thus  causing  the  speed  of 
the  engine  to  decrease,  upon  which  the  reverse  series  of  opera- 
tions takes  place  and  the  valve  opens  again.  It  is,  of  course,  the 
purpose  of  the  device  to  find  such  a  position  for  the  valve  V  that 
it  will  just  keep  the  engine  running  at  uniform  speed,  by  admit- 
ting just  the  right  quantity  of  steam  for  this  purpose. 

178.  Theory   of    Governor. — Several   different    forms   of   the 
governor  are  shown  later  in  the  present  chapter  and  will  be  dis- 


FIG.  123. 

cussed  subsequently,  but  it  may  be  well  to  begin  with  the  simplest 
form  shown  in  Fig.  123,  where  the  connection  of  the  sleeve  to 
the  valve  is  not  so  direct  as  in  Fig.  122  but  must  be  made  through 
suitable  linkage.  The  left-hand  figure  shows  a  governor  with  the 
arms  pivoted  on  the  spindle,  while  the  right-hand  figure  shows 
the  pivots  away  from  the  spindle,  and  the  same  letters  are  used 
on  both.  Let  the  total  weight  of  the  two  balls  be  w  lb.,  each  ball 

therefore  weighing  -^  lb.,  and  let  these  be  rotated  in  a  circle  of 

Zi 

radius  r  ft.,  the  spindle  turning  at  n  revolutions  per  minute  corre- 

27T72. 

spending  to  co  =  -^=r  radians  per  second.  For  the  present, 
friction  will  be  neglected. 


GOVERNORS  205 

Three  forces  act  upon  each  ball  and  determine  its  position  of 
equilibrium.  These  are:  (a)  The  attraction  of  gravity,  which  will 
act  vertically  downward  and  will  therefore  be  parallel  with  the 
spindle  in  a  governor  where  the  spindle  is  vertical  as  in  the  illus- 

nij 

tration  shown.     The  magnitude  of  this  force  is  ^  pds.     (6)  The 

£t 

second  force  is  due  to  the  centrifugal  effect  and  acts  radially 
and  at  right  angles  to  the  spindle,  its  amount  being  - -— .  r.  co2  pds. 

t7 

(c)  The  third  force  is  due  to  the  pull  of  the  ball  arm,  and  will  be 
in  the  direction  of  the  line  joining  the  center  of  gravity  of  the 
ball  to  the  pivot  on  the  spindle,  which  direction  may  be  briefly 
called  the  direction  of  the  ball  arm. 

These  three  forces  must  be  in  equilibrium  so  that  the  vector 

triangle  ABC  may  be  drawn  where  AB  =  „>  BC  —  o~rc°2  anc* 

co  must  be  such  that  AC  is  parallel  to  the  arm.  Now  let  D  be 
the  point  at  which  the  ball  arm  intersects  the  spindle  and  draw 
AE  perpendicular  to  the  spindle  DE;  then  AE  =  r,  the  radius 
of  rotation  of  the  balls  and  the  distance  DE  =  h  is  called  the 
height  of  the  governor. 

The  triangles  DAE  and  ACB  are  similar  and  therefore: 

DE  _  AB 
EA  ~  BC 
or 

w 

h  _        2 
r  ~~  w 

^ 
which  gives 

g 

h  =  ~2 
to* 

Thus,  the  height  of  the  governor  depends  on  the  speed  alone  and 
not  on  the  weight  of  the  balls.  The  investigation  assumes  that 
the^  resistance  offered  at  the  sleeve  is  negligible  as  indeed  is  the 
case  with  many  governors  and  gears,  but  allowance  will  be  made, 
for  this  in  problems  discussed  later. 

179.  Defects  of  this  Governor. — Such  a  governor  possesses 
several  serious  defects.  In  the  first  place,  the  sleeve  must  move 
in  order  that  the  valve  may  be  operated,  and  this  movement  of 
the  sleeve  will  evidently  correspond  with  a  change  in  the  height 


206  THE  THEORY  OF  MACHINES 

and  hence  with  a  change  in  speed  co.  Thus,  each  position  of  the 
balls,  corresponding  to  a  given  valve  position,  means  a  different 
speed  of  the  governor  and  therefore  of  the  engine ;  this  is  what  the 
governor  tries  to  prevent,  for  its  purpose  is  to  keep  the  speed  of 
the  engine  constant,  although  the  valve  may  have  to  be  opened 
various  amounts  corresponding  to  the  load  which  the  engine 
carries.  This  defect  may  be  briefly  expressed  by  saying  that  the 
governor  is  not  isochronous,  the  meaning  of  isochronism  being 
that  the  speed  of  the  governor  will  not  vary  during  the  entire 
range  of  travel  of  the  sleeve,  or  in  other  words  the  valve  may  be 
moved  into  any  position  to  suit  the  load,  and  yet  the  engine  and 
therefore  the  governor,  will  always  run  at  the  same  speed. 

The  second  defect  is  that  for  any 
reasonable  speed  h  is  extremely  small. 
To  show  this  let  the  governor  run  at 
120  revolutions  per  minute  so  that  co 

=  -~Q-  =   12.57   radians  per   second; 
then  h  =  ~  =  ,,0'r^o  =  0.2036ft. or 


2.44  in.,  a  dimension  which  is  so  small, 
that  if  the  balls  were  of  any  reasonable 
size,  it  would  make  the  practical  con- 
struction almost  impossible. 
FIG.  124.— Crossed-arm      180.  Crossed -arm  Governor. — Now 

it  is  the  desire  of  all  builders  to  make 

their  governors  as  nearly  isochronous  as  is  consistent  with  other 
desirable  characteristics,  which  means  that  the  height  h  must  be 
constant,  and  to  serve  this  end  the  crossed-arm  governor  shown 
in  Fig.  124,  has  been  built  somewhat  extensively.  The  propor- 
tions which  will  produce  isochronism  may  be  found  mathematic- 
ally thus: 

Inspection  of  the  figure  shows  that 

h  =  I  cos  6  —  a  cot  6. 

For  isochronism  h  is  to  remain  constant  for  changes  in  the 
angle  6  or 

-77  =  0  =  —  Z  sin  0  +  a  cosec2  8. 
do 

From  which 

a  =  I  sin3  B 
h  =  I  cos3  0 


i  -   3-  "TV  e/ 

h  -  ((,  -    t---  c        LA 

GOVERNORS  207 

and  therefore  a  =  Zsin3  6  =  h  tan3Q.  =—2  tan2  0;  which  formulas 

give  the  relations  between  a,  Z  and  0,  and  it  will  be  noticed  that 
the  weight  w  does  not  enter  into  the  calculation  any  more  than  it 
does  into  the  time  of  swing  of  the  pendulum. 

As  an  example  let  the  speed  be  w  =  10  radians  per  second 
(corresponding  to  97  revolutions  per  minute)  and  let  6  =  30°. 
Then  the  formulas  give  a  =  0.0618  ft.  or  0.74  in.,  I  =  0.495  ft. 
or  5.94  in.  and  the  value  of  h  corresponding  to  6  =  30°  is  0.322 
ft.  With  these  proportions  the  value  of  h  when  8  becomes  35° 
will  be  0.317  ft.,  a  decrease  of  1.56  per  cent.,  corresponding  to  a 
change  of  speed  of  about  0.8  per  cent. 

With  a  governor  as  shown  at  Fig.  123  and  co  =  10  as  before,  a 
change  from  30°  to  35°  produces  a  change  in  speed  of  about  3  per 
cent. 

It  is  possible  to  design  a  governor  of  this  type  which  will 
maintain  absolutely  constant  speed  for  all  positions  of  the  balls, 
and  the  reader  may  prove  that  for  this  it  is  only  necessary  to  do 
away  with  the  ball  arms,  and  place  the  balls  on  a  curved  track 
of  parabolic  form,  so  that  they  will  always  remain  on  the  surface 
of  a  paraboloid  of  revolution  of  which  the  spindle  is  the  axis. 
In  such  a  case,  h  and  therefore  u  will  remain  constant. 

A  perfectly  isochronous  governor,  however,  has  the  serious 
defect  that  it  is  unstable  or  has  no  definite  position  for  a  given 
speed,  and  thus  the  slightest  disturbing  force  will  cause  the  balls 
to  move  to  one  end  or  other  of  their  extreme  range  and  the  gov- 
ernor will  hunt  for  a  position  where  it  will  finally  come  to  rest. 
Such  a  condition  of  instability  is  not  admissible  in  practice  and 
designers  always  must  sacrifice  isochronism  to  some  extent  to 
the  very  necessary  feature  of  stability,  because  the  hunting  of 
the  balls  in  and  out  for  their  final  position  means  that  the  valve 
is  being  opened  and  closed  too  much  and  hence  the  engine  is 
changing  its  speed  continually,  or  is  racing.  In  the  simple 
governor  quoted  in  Sec.  178  it  is  evident  that  while  it  is  not 
isochronous  it  is  stable,  for  each  position  of  the  balls  corresponds 
to  a  different  but  definite  speed  belonging  to  the  corresponding 
value  of  the  height  h. 

181.  Weighted  or  Porter  Governor. — In  order  to  obviate 
these  difficulties  Charles  T.  Porter  conceived  the  idea  of  plac- 
ing on  the  sleeve  a  heavy  central  weight,  free  to  move  up  and 
down  on  the  spindle  and  having  its  center  of  gravity  on  the 


208 


THE  THEORY  OF  MACHINES 


axis  of  rotation.  This  modified  governor  is  shown  in  Fig.  125, 
with  the  arms  pivoted  on  the  spindle,  although  sometimes  the 
arms  are  crossed  and  when  not  crossed  they  are  frequently  sus- 


FIG.  125. — Porter  governor. 

pended  by  pins  not  on  the  spindle.     In  Fig.  126  a  similar  gover- 
nor is  shown  diagrammatically,  with  the  pivots  0  to  one  side. 
To  study  the  conditions  of  equilibrium  of  such  a  governor 
find  the  image  Q'  of  the  point  Q  where  the  link  Z2  is  attached  to 


GOVERNORS 


209 


the  central  weight  W.  Then  by  the  propositions  of  Chapter  IX 
the  half  of  the  weight  which  acts  at  Q  may  be  transferred  to  Q'  , 
and  let  it  be  assumed  that  l\  and  Z2  are  of  equal  length;  then  by 
taking  moments  of  the  weights  and  centrifugal  force  about  0 
the  equation  is 


W 

-77-  •  21  1  sin 


w  w 

-oh  sin  B  —  -^- 
z  zg 


cos 


From  which  it  follows  that 
h  = 


FIG.  126. — Porter  or  weighted  governor. 

For  example,  let  li  =  Z2  =  9  in.  or  0.75  ft.,  speed  194  revolutions' 
per  minute  for  which  w  =  20  radians-  per  second,  and  let  each  ball 
weigh  4  lb.,  i.e.,  w  =  8  lb.,  6  =  45°  and  ai  =  a2  =  0.  Then 
by  measuring  from  a  drawing,  or  by  computation,  h  is  found  to 
be  0.53  ft.,  and 


w 


gives 


or 


2W  +  w 


wh~  =  8  X  0.53  X 

Q 

W  =  22.4  lb. 


=  52.8  lb. 


182.  Advantages  of  Weighted  Governors. — The  first  advantage 
of  such  a  governor  is  that  the  height  h  may  be  varied  within 

14 


210  THE  THEORY  OF  MACHINES 

wide  limits  at  any  given  speed  by  a  change  in  the  central  weight 
W,  and  thus  the  designer  is  left  much  freedom  in  proportioning 
the  parts.  In  the  numerical  example  above  quoted  the  height 
would  be  6.6  times  as  great  as  for  an  unweighted  governor  run- 
ning at  the  same  speed,  since  -  -  =  6.6. 

Again  the  variation  in  height  h  corresponding  to  a  given  change 
in  speed  is  much  increased  by  the  use  of  the  central  load,  with  the 
result  that  the  sleeve  will  move  through  a  certain  height  with 
smaller  change  in  speed.  Now  the  travel  of  the  sleeve,  or  the 
lift  as  it  is  often  called,  is  fixed  by  the  valve  and  its  mechanism, 
and  the  above  statement  means  that  for  a  given  lift  the  variation 
in  speed  will  be  decreased,  or  the  governor  will  become  more 
sensitive.  By  sensitiveness  is  meant  the  proportional  change 
of  speed  that  occurs  while  the  sleeve  goes  through  its  complete 
travel,  the  governor  being  most  sensitive  which  has  the  least 
variation. 

To  prove  this  property  let  h',  h,  co'  and  co  represent  the  heights 
and  speeds  corresponding  to  the  highest  and  lowest  positions 
of  the  sleeve. 
Then 

_£_  and 


W  r  W  co" 

CO 

or 

k'  /C0\2  CO 

T~   =  (~7)         or       ~~/   = 

But  since  h  and  In'  are  much  greater  in  the  weighted  than  in 
the  unweighted  governor,  therefore  — ,  is  more  nearly  unity  in  the 

former  case. 
Again, 

'2  7  /2  2  i,/         ft 

CO  Al  CO        —    CO  /i     —   fl 


CO2     =~    h'  CO2  "     h' 

Therefore, 

co'  —  co      co'  +  co  h'   —  h 

co  co  hr 

Now  usually  co'  and  co  do  not  differ  very  much,  so  that  co'  +  o>  = 
2co  nearly,  and  therefore, 


GOVERNORS  211 

-  co  h'-  h 


The  relation  —  is  evidently  the  sensitiveness  of  the  governor1 

and  the  smaller  the  ratio  the  more  sensitive  is  the  governor. 
For  an  isochronous  governor  6co  =  0. 

To  compare  the  weighted  and  unweighted  governors  in  regard 
to  sensitiveness  take  the  angular  velocity  co  =  10  radians  per 
second  and  let  W  =  60  Ib.  and  fjfa  =  8  Ib.  Let  the  change  in 
height  necessary  to  move  the  sleeve  through  its  entire  lif^Tbe 
Kin. 

(a)  Unweighted  Governor.  —  For  the  data  given  h  =  3.86  in., 
and,  therefore,2 

dh        0.5 


h  '    3.86 


=  0.129. 


Hence,  2—  =  0.129  or  —  =  0.064  or  6.4  per  cent.,  so  that  the 

CO  CO 

variation  in  speed  will  be  6.4  per  cent. 

(6)  Weighted  Governor. — For  this  governor 


11  — 

-  A  o.ou  — 

A 

W 

8 

and 

dh 

0.5 

0.008 

h  ~ 

61.76 

or 

£  JJ 

2—  =  0.008  giving  —  =  0.004  or  0.4  per  cent. 

CO  CO 

the  variation  in  speed  being  only  0.4  per  cent.     Such  a  gover- 
nor would  therefore  be  very  nearly  isochronous. 

A  third  property  of  this  weighted  governor  is  that  it  is  power  - 

1  This  may  be  simply  shown  by  the  calculus  thus : 

,   _  2W  +  w     g_ 

W  '    «» 

and  differentiating, 

2W  -f  w  8h  n  8  a) 

8h  = -  •  2W.  So,.      .  .  -r  =   —  2  — 

w  h  w 

or 

5co  dh 

2  ~^  =     "  h 

where  5o>  and  8h  represent  the  small  changes  taking  place  in  to  and  h. 

2  The  negative  sign  appearing  before  dh  on  the  preceding  formula  merely 
means  that  an  increase  in  speed  corresponds  to  a  decrease  in  height. 


212  THE  THEORY  OF  MACHINES 

ful,  that  is  to  say,  that  as  the  central  weight  is  very  heavy  the 
equilibrium  of  the  device  is  very  little  affected  by  any  slight  dis- 
turbing force,  such  as  that  required  to  operate  the  valve  gear  or 
to  overcome  friction.  Powerfulness  is  a  very  desirable  feature, 
for  it  is  well  known  in  practice  that  the  force  required  to  operate 
the  valve  gear  is  not  constant  and  therefore  produces  a  variable 
effect  on  the  governor  mechanism,  which,  unless  the  governor  is 
powerful,  is  sufficient  to  move  the  weights,  causing  hunting. 

The  Porter  governor  thus  enables  the  designer  to  make  a  very 
sensitive  governor,  of  practical  proportions  and  one  which  may 
be  made  as  powerful  as  desired,  so  that  it  will  not  easily  be 
disturbed  by  outside  forces. 

THE  CHARACTERISTIC  CURVE 

183.  A  number  of  the  results  and  properties  of  governors 
may  be  graphically  represented  by  means  of  characteristic  curves, 
and  it  will  be  convenient  at  this  stage  to  explain  these  curves  in 
connection  with  the  Porter  governor.  Let  Fig.  127  (a)  represent 
the  right-hand  part  of  a  Porter  governor,  the  letters  having  the 
same  significance  as  before.  Choose  a  pair  of  axes,  OC  in  the 
direction  of  the  spindle  and  OA  at  right  angles  to  the  spindle, 
and  let  the  centrifugal  force  on  the  ball  be  plotted  vertically 
along  OC,  as  against  radii  of  rotation  of  the  balls,  which  are 
plotted  along  OA,  r±  and  r%  representing  respectively  the  inner 
and  outer  limiting  radii,  the  resulting  figure  will  usually  be  a 
curved  line  somewhat  similar  to  CiCCz  in  Fig.  127 (a). 

Let  the  angular  velocities  corresponding  to  the  radii  r\  and  r2 
be  coi  and  co2  radians  per  second  respectively,  and  let  co  = 
J£  (^i  +  W2)  represent  the  mean  angular  velocity  to  which  the 
corresponding  radius  of  rotation  is  r  ft. 

Then 


W  2 

-   TZ    C02 


and 


g 

where  the  forces  C,  C\  and  Cz  are  the  total  centrifugal  forces 
acting  on  the  two  balls.  The  properties  of  this  curve,  which 
may  be  briefly  called  the  C  curve,  may  now  be  discussed. 


GOVERNORS 


213 


1.  Condition  for  Isochronism. — If  the  governor  is  to  be  iso- 
chronous then  the  angular  velocity  for  all  positions  of  the  balls 
must  be  the  same,  that  is  co  =  coi  =  402  and  hence  the  centrifugal 
force  depends  only  on  the  radius  of  rotation  (see  formulas  above) 
or 

Ci      Cz      C 

—  =  —  =  —  =  a  constant, 

7i  7  2  * 

a  condition  which  is  fulfilled  by  a  C  curve  forming  part  of  a 
straight  line  passing  through  0.  Thus  any  part  of  OC  would 
satisfy  this  condition  and  the  part  ED  corresponds  to  the  radii 
TI  and  r2  in  the  governor  selected. 


Feet 


(a) 
FIG.  127. — Characteristic  curve. 


2.  Condition  for  Stability. — Although  the  curve  ED  will  give 
an  isochronous  governor,  it  produces  instability.  The  curve 
CiCCz  indicates  that  the  speeds  are  not  the  same  for  the  various 
positions  of  the  balls,  and  a  little  consideration  will  show  that 
Ci  corresponds  to  a  lower  speed  and  Cz  to  a  higher  speed  than  C. 
This  is  evident  on  examining  the  conditions  at  radius  n,  for  the 
point  E  corresponds  to  the  same  speed  as  C,  but  since  E  and  C\ 
are  both  taken  at  the  same  radius,  and  since  the  centrifugal  force 
FE  is  greater  than  FC\  it  is  evident  that  the  angular  velocity  cor 
corresponding  to  C\  is  less  than  the  angular  velocity  co  correspond- 


214  THE  THEORY  OF  MACHINES 

ing  to  E.  Thus  a  curve  such  as  dCC2,  which  is  steeper  than  the 
isochronous  curve  where  they  cross,  indicates  that  the  speed 
of  the  governor  will  increase  when  the  balls  move  out,  and  it  may 
similarly  be  shown  that  such  a  curve  as  Ci'CCz',  which  is  flatter 
than  the  isochronous  curve,  shows  that  the  speed  of  the  governor 
decreases  as  the  balls  move  out. 

Now  an  examination  of  these  curves  shows  that  the  one  CiCC2 
belongs  to  a  governor  that  is  stable,  for  the  reason  that  when  the 
ball  is  at  radius  ri  it  has  a  definite  speed  and  in  order  to  make  it 
move  further  out  the  centrifugal  force  must  increase.  But 
on  account  of  the  nature  of  the  curve  the  centrifugal  force  must 
increase  faster  than  the  radius  or  the  speed  must  increase  as  the 
ball  moves  out,  and  thus  to  each  radius  there  is  a  corresponding 
speed.  On  the  other  hand,  the  curve  Ci'CCz  shows  an  entirely 
different  state  of  affairs,  for  at  the  radius  r\  the  centrifugal  force 
is  greater  than  FE  or  the  ball  has  a  higher  speed  than  co  and  thus 
as  the  ball  moves  out  the  speed  will  decrease.  Any  force  that 
would  disturb  the  governor  would  cause  the  ball  to  fly  outward 
under  the  action  of  a  resultant  force  Ci'E,  and  if  it  were  at  radius 
r2  any  disturbance  would  cause  the  ball  to  move  inward. 

Another  way  of  treating  this  is  that  for  the  curve  CiCCz  the 
energy  of  the  ball  due  to  the  centrifugal  force  is  increasing  due 
both  to  the  increase  in  r  and  in  the  speed,  and  as  the  weights 
W  and  w  are  being  lifted,  the  forces  balance  one  another  and  there 
is  equilibrium;  whereas  with  the  curve  Ci'CCz  there  is  a  decrease 
in  speed  and  also  in  the  energy  of  the  balls  while  the  weights  are 
being  lifted  and  the  forces  are  therefore  unbalanced  and  the 
governor  is  unstable. 

Thus,  for  stability  the  C  curve  must  be  steeper  than  the  line 
joining  any  point  on  it  to  the  origin  0.  Sometimes  governors 
have  curves  such  as  those  shown  at  Fig.  127(6)  and  curve  CiCCa 
indicates  a  stable  governor,  d'CC2'  an  unstable  governor, 
CiCCz  partly  stable  and  partly  unstable  and  finally  CYCCj 
partly  unstable  and  partly  stable. 

3.  Sensitiveness. — The  shape  of  the  curve  is  a  measure  of 
the  sensitiveness  of  the  governor.  If  S  indicates  the  sensitive- 

Ci?2     —     Wl 

ness,  then  by  definition  S  =  - 
Now 

2    ~          2 


a>2  —  coi  _  yo?2  —  C01JICQ2  ~t~  ^L)  _  002 
co  co(co2  ~T"  ^i)  2co 


GOVERNORS 


215 


since   co2  +  coi  =  2co  nearly. 
Therefore 


S  = 


1        O>22     — 


CO' 


But 
and 
and 


w 


2    == 


d  =  - 


=  —  ror. 


Hence,  by  substituting  in  the  formula  for  S,  the  result  is 


w          w 

rc2    di 

.  A*                       .     rs* 

g    '        g 

1 

r2        7*1 

C 

~  2 

C 

w 

r 

Q    _        "       y  v 

O      c\  /^i  ~     c\  S~i 


Referring  now  to  Fig.  127  it  is  seen  that 


and 


-  -  " 


C  DA 

7  ==  tan  ^  =  OA 


or 


S 


tan 


2L 


tar0 


fi  _ 
J  ~ 


-  tan  0fi  _  1  rC2A  -  JgAi  _  1  (V? 
~  ^    ~  2  DA  ' 


2          DA 


Thus  the  C  curve  is  also  valuable  in  showing  the  sensitiveness 
of  the  governor.  For  an  isochronous  governor  C2,  B  and  D 
coincide  and  S  =  0.  Evidently  the  more  stable  the  governor  is 
the  less  sensitive  it  is,  and  in  a  general  way  an  unstable  governor 
is  more  sensitive  than  a  stable  one.  At  d,  Fig.  127 (a),  the  stable 
governor  is  most  nearly  isochronous,  arid  evidently  a  fair  degree 
of  stability  and  sensitiveness  could  both  be  obtained  in  a  governor 
having  a  reverse  curve  with  point  of  inflexion  near  C,  the  part 
CCZ  being  concave  to  OA,  the  part  CiC  convex  to  OA. 

4.  Powerfulness. — The  C  curve  also  shows  the  powerfulness 
of  the  governor,  since  in  this  curve  vertical  distances  represent 


216 


THE  THEORY  OF  MACHINES 


the  centrifugal  forces  acting  on  the  balls,  while  horizontal  distances 
represent  the  number  of  feet  the  balls  move  horizontally  in  the 
direction  of  the  forces.  Thus,  an  elementary  area  represents  the 
product  C.dr  ft.-pds.  and  the  whole  area  between  the  C  curve  and 
the  axis  OA  gives  the  work  done  by  the  balls  in  moving  over 
their  entire  range,  and  is  therefore  the  work  available  to  move 
the  valve  gear  and  raise  the  weights.  The  higher  the  curve  is 
above  OA  the  greater  is  the  available  work,  and  this  clearly  cor- 
responds to  increased  speed  in  a  given  governor. 

5.  Friction. — The  effect  of  friction  has  been  discussed  in  the 
previous  chapter  and  need  not  be  considered  here.  Some  writers 
treat  friction  as  the  equivalent  of  an  alteration  to  the  central 
weight,  and  if  this  is  done  the  effect  is  very  well  shown  in  Fig.  128 
where  the  C  curve  for  the  frictionless  governor  is  shown  at 

CiCC2.     As  the  weight  W  is  lifted 
4        the  effect  of  friction  when  treated 
c2     in  this  way  is  to  increase  W  by 
e        the  friction/  with  the  result  that 
the    C   curve  is  raised  to  3-4, 
whereas  when  the  weight  W  is 
falling  the  friction  has  the  effect 
of  decreasing  the  weight  W  and 
JS*"   to  lower  the  C  curve  to  5-6.     The 
FIG.  128.  effects    of    these    changes    are 

evident  without  discussion. 

184.  Relative  Effects  of  the  Weights  of  the  Balls  and  the 
Central  Weight. — For  the  purpose  of  further  understanding  the 
governor  and  also  for  the  purpose  of  design,  it  is  necessary  to 
analyze  the  effects  of  the  weights  separately.  Referring  to  Fig. 
129  and  finding  the  phorograph  by  the  principles  of  Chapter  IV 
the  image  of  D  is  at  D'  and  taking  moments  about  A,  remember- 

W 

ing  that  •„-  may  be  transferred  from  D  to  its  image  D', 

MWb  +  M™  -MCh  =  0. 

(Sec.  151,  Chapter  IX).  Now  let  Cw  be  the  part  of  C  neces- 
sary to  support  W  and  Cw  the  corresponding  quantity  for  w,  so 
that 


Cw  +  Cw  =  C  where  C  =  - 


But 


GOVERNORS 


217 


That  is 


b  e 

W  T  and  Cw  =  w-r- 

n  h 


The  graphical  construction  is  shown  in  Fig.  129.  Draw  JH 
and  LG  horizontally  at  distances  below  A  to  represent  w  and  W 
respectively,  then  join  AE,  the  line  D'E  being  a  vertical  through 
Dr.  Then  it  may  be  easily  shown  that 

Cw  =  AF    and  Cw  =  AK. 


•'i 

FIG.  129. — Governor  analysis. 

Making  this  construction  for  various  positions  and  plotting 
for  the  complete  travel  of  the  balls  the  two  curves  are  as  drawn 
in  Fig.  129. 

185.  Example. — The  following  dimensions  are  taken  from  an 
actual  governor  and  refer  to  Fig.  129.  ai  =  0,  az  =  1J^  in.,  AB  = 
12K  in.,  AM  =  16  in.  and  BD  =  lOJ^  in->  while  the  travel 
of  the  sleeve  is  2J^  in.  and  the  point  D  is  15%  in.  below  A 
when  the  sleeve  is  at  the  top  of  its  travel.  Each  ball  weighs 
15  Ib.  so  that  w  =  30  lb.,  also  W  =  124  Ib. 

Then  drawing  the  governor  mechanism  in  the  upper,  the  mean 


218 


THE  THEORY  OF  MACHINES 


and  the  lowest  positions  of  the  sleeve,  the  following  table  of 
results  is  obtained,  since  for  the  ball  -  =  00^,  ~  =  0.933. 


RESULTS  ON  PORTER  GOVERNOR 


Sleeve 
position 

& 

feet 

r  =  e 
feet 

h 
feet 

wl  =  cw 

ft 

pounds 

w\  =  Cw 
h 
pounds 

Cw  +  Cw  =  C 
pounds 

W2  =  -^ 

w 
—  r 
0 

n 
rev.  per 
min. 

1.  Upper.  . 

1.51 

0.98 

0.90 

208.2 

32.8 

241.0 

263.7 

155.2 

2.  Mean... 

1.40 

0.92 

0.97 

178.3 

27.5 

206.8 

241.3 

148.5 

3.  Lower... 

1.25 

0.83 

1.04 

149.1 

24.0 

173.1 

222.8 

142.3 

The  corresponding  C  curve  and  the  two  components  Cw  and 
Cw  are  plotted  at  Fig.  130  from  which  it  is  clear  that  the  governor 


FIG.  130. 

is  stable.     From  this   curve  it  appears   that  the  sensitiveness 

.      ,/        37 

is    %  X  222  =  0-^856    or   8.56   per    cent.,  which    checks  very 

well  with  the  speeds  as  shown  in  the  last  column,  and  which  indi- 
cates a  sensitiveness  of  8.65  per  cent. 

If  it  is  desired  to  find  the  position  of  the  balls  for  a  speed  of 
150  revolutions  per  minute,  then  o>  =  15.7  radians  per  second 

IV 

and  the  force  C  =  —  rco2  =  0.933  X  r  X  246.5  =  230  r.     Then 
y 

C 
draw  the  line  for  which  the  tangent  is  —  =  230  and  where  it  cuts 

the  C  curve  is  the  radius  of  the  balls  corresponding  to  this 
speed. 

Assuming  the  mean  height  of  the  C  curve  to  be  207  pds.  the 
work  done  in  the  entire  travel  of  the  balls  is  207  (0.98  —  0.83)  = 
31  ft.-pds. 


GOVERNORS 


219 


2  =  10  in.,  and  BM  =  3 


13.6 
R 


186.  Design  of  a  Porter  Governor. — These  curves  may  be  con- 
veniently used  in  the  design  of  a  Porter  governor  to  satisfy  given 
conditions.  Let  it  be  required  to  design  a  governor  of  this  type 
to  run  at  a  mean  speed  of  200  revolutions  per  minute  with  a 
possible  variation  of  less  than  5  per  cent,  either  way  for  the  ex- 
treme range.  The  sleeve  is  to  have  a  travel  of  2  in.  and  the 
governor  is  to  have  a  powerfulness  represented  by  20  ft.-pds. 

From  general  experience  select  the  dimensions  a\,  «2,  li,  Z2 
and  BM  in  Fig.  129.  Thus  take 
in. ;  also  make  «i  =  a2  =  1  in. 
Draw  the  governor  in  the  central 
position  of  the  sleeve  with  the 
arms  at  90°,  as  this  angle  gives 
greater  uniformity  than  other 
angles,  and  measure  the  extreme 
radii  and  also  that  for  the  central  origin 
position  of  the  sleeve.  The  C 
curve  may  now  be  constructed 
and  at  Fig.  131  the  three  radii 
are  marked,  which  are  r\  =  9.5 
in.,  r  =  10.22  in.  and  r2  =  10.82 
in.  Now  the  power  of  the 
governor  is  20  ft.-pds.,  and  divid- 
ingthisby  r2  —  7*1  =  O.llft.  gives 
the  mean  height  of  the  C  curve 
as  182  pds.  Plot  this  at  radius  r 
making  HG  =  182  pds.  and  join 
to  0;  it  cuts  r2  at  D. 

Now  the  sensitiveness  is  to  be 
5  per  cent.,  so  that  T  and  U  are 
found  such  that  DT  =  DU  = 
0.05  X  AD  =  9.65  pds.  Join 
T  and  U  to  0,  thus  locating  V  and  the  resulting  C  curve  will 
be  VGT  shown  dotted. 

Next,  since  the  centrifugal  force  GH  =  182  pds.  corresponds  to 
a  radius  r  =  10.22  in.  and  a  speed  of  200  revolutions,  the  weight 

w  may  be  found  from  the  formula  C  =  —  rco2  and    gives  w  = 

\y 

15.75  Ib.  By  the  use  of  such  a  diagram  as  Fig.  129  the  three 
values  of  Cw  are  measured  for  the  three  radii  and  the  Cw  curve  is 
drawn  in  Fig.  131,  and  then  the  values  of  Cw  are  found.  Thus, 


15.8 


Curve 
H 


-160 


-140 


-120 


-100 


-  40 


-  20 
18.1 


rj-9.5"    r  =  10.22"  r2  =  10.82" 

FIG.  131. — Governor  design. 


220 


THE  THEORY  OF  MACHINES 


RV scales  off  as  153  pds.  and  hence  CWl  =  153  -  13.6=  139.4  pds., 
and  similarly  the  other  values  of  CV  are  found,  and  from 
them  W i  =  104.6  pds.,  W  =  107.6  pds.  and  W2  =  110.6  pds.  are 
obtained,  as  shown  at  Fig.  129. 

As  a  trial  assume  the  mean  of  these  values  W  —  107.6  pds.,  as 
the  value  of  the  central  weight  and  proceeding  as  in  Sec.  185  find 
the  three  new  values  of  Cw  and  also  of  C  =  Cw  +  Cw  and  lay 
these  off  at  the  various  radii  giving  the  plain  curve  in  Fig.  131. 
This  will  be  found  to  correspond  to  a  range  in  speeds  from  192 


FIG.   132. — Proell  governor. 

to  207.5  revolutions  per  minute,  and  as  this  gives  less  than  a 
5  per  cent,  variation  either  way  from  the  mean  speed  of  200 
revolutions  it  would  usually  be  satisfactory.  If  it  is  desired  to 
have  the  exact  value  of  5  per  cent.,  then  it  will  be  better  to  start 
with  a  little  larger  variation  of  say  6  per  cent,  and  proceed  as 
above. 

187.  Proell  Governor. — The  method  already  described  may 
be  applied  to  more  complicated  forms  of  governor  with  the  same 
ease  as  is  used  in  the  Porter  governor,  the  phorograph  making 


GOVERNORS  221 

these  cases  quite  simple.  As  an  illustration,  the  Proell  governor 
is  shown  in  Fig.  132  and  is  similarly  lettered  to  Fig.  129,  the 
difference  between  these  governors  being  that  in  the  Proell  the 
ball  is  fastened  to  an  extension  of  the  lower  arm  DB  instead  of 
the  upper  arm  AB  as  in  the  Porter  governor. 

As  before,  AB  is  chosen  as  the  link  of  reference  and  the*  images 
found  on  it  of  the  points  D  and  M  by  the  phorograph,  Chapter  IV. 
The  force  }^W  is  then  transferred  to  Df  and  J^C  and  %w  to  Mf 
from  Chapter  IX,  but  in  computing  C  the  radius  is  to  be  measured 
from  the  spindle  to  M  and  not  to  Mf,  since  the  former  is  the 
radius  of  rotation  of  the  ball.  The  meanings  of  the  letters  will 
appear  from  the  figure  and  by  taking  moments  about  A  the  same 
relation  is  found  as  in  Sec.  184.  The  results  for  the  complete 
travel  of  the  balls  is  shown  on  the  lower  part  of  Fig.  132. 

SPRING  GOVERNORS 

188.  Spring  governors  have  been  made  in  order  to  eliminate 
the  central  weight  and  to  make  possible  the  use  of  a  nearly 
isochronous  and  yet  sensitive  and  powerful  governor.     These 
governors  always  run  at  high  speed  and  are  sometimes  mounted 
on  the  main  engine  shaft,  but  more  frequently  on  a  separate 
spindle. 

189.  Analysis  of  Hartnell  Governor. — One  form  of  this  gover- 
nor, frequently  ascribed  to  Hartnell  of  England,  is  shown  in  Fig. 
133  and  the  action  of  the  governor  may  now  be  analyzed.     Let 
the  total  weight  of  the  two  balls  be  w  lb.,  as  before,  and  let  W 
denote  the  force  on  the  ball  arms  at  BB,  due  to  the  weight  of  the 
central  spring  and  any  additional  weight  of  valve  gear,  etc.     In 
this  case  W  will  remain  constant  as  in  the  loaded    governor. 
Now  let  F  be  the  pressure  produced  at  the  points  B  by  the 
spring,  F  clearly  increasing  as  the  spring  is  compressed  due  to 
the  outward  motion  of  the  balls. 

In  dealing  with  governors  of  this  class  it  is  best  to  use  the  mo- 
ments of  the  forces  about  the  pin  A  in  preference  to  the  forces 
themselves,  and  hence  in  place  of  a  C  curve  for  this  governor  a 
moment  or  M  curve  will  be  plotted  in  its  place,  the  radius  of 
rotation  of  the  balls  being  used  as  the  horizontal  axis.  The 
symbols  M ,  MF,  Mw  and  M w  indicate,  respectively,  the  mo- 
ments about  A  of  the  centrifugal  force  C,  the  spring  force  F, 
the  weight  of  the  balls  w  and  the  dead  weight  W  along  the  spindle. 
Then  M  =  MF  +  M w  +  Mw 


222 


THE  THEORY  OF  MACHINES 


or  Ca  cos  6  =  Fb  cos  6  +  Wb  cos  6  —  wa  sin  6. 

The  moment  curves  may  be  drawn  and  take  the  general  shapes 
shown  in  Fig.  133  and  similar  statements  may  be  made  about 
these  curves  as  about  those  for  the  Porter  governor.  If  it  is 
desired,  the  corresponding  C  curves  may  readily  be  drawn 
from  the  formula 


or 


Ca  cos  6  =  Fb  cos  6  +  Wb  cos  0  —  wa  sin  0 

C 

—  CF  +  Cw 


p  -  _]_  iff w  tan 

a  a 


FIG.  133. — Hartnell  governor. 

and  a  graphical  method  for  finding  these  values  is  easily  devised. 
The  curve  for  W  is  evidently  a  horizontal  line  since  W,  b  and  a 
are  all  constant,  while  that  for  w  is  a  sloping  line  cutting  the 
axis  of  r  under  the  pin  A  and  the  CF  curve  may  be  found  by 
differences. 

190.  Design  of  Spring. — The  data  for  the  design  of  the  spring 
may  be  worked  out  from  the  CF  curve  found  as  above.     Evidently 

Cp  =  F~  or  F  =  CF  X  r-  =  CF  X  a  constant,  and  thus  from 
the  curve  for  CF  it  is  possible  to  read  forces  F  to  a  suitable  scale. 


GOVERNORS 


223 


These  forces  F  may  now  be  plotted  as  at  Fig.  134  which  gives 
the  values  of  F  for  the  different  radii  of  rotation.  As  the  line 
EGL  thus  found  is  slightly  curved,  no  spring  could  exactly  fulfil 
the  requirements,  but  by  joining  E  and  L  and  producing  to  H,  a 
solution  may  be  found  which  will  fit  two  points,  E  and  L,  and  will 
nearly  satisfy  other  points.  Draw  EK  horizontally;  then  LK 
represents  the  increase  in  pressure  due  to  the  spring  while  the 
balls  move  out  EK  in.,  and  hence  the  spring  must  be  such  that 
T  JC 

r  pds.  will  compress  it  1  in.,  and  further  the  force  produced  by 


Feet 


the  spring  when  the  balls  are  in  is  EJ  pds.,  that  is  the  spring 
must  be  compressed  through  HJ  in.,  for  the  inner  position  of 
the  balls. 

In  ordinary  problems  it  is  safe  to  assume  for  preliminary  cal- 
culations that  the  effect  of  the  weights  W  and  w  can  be  neglected 
and  the  spring  may  be  designed  to  balance  the  centrifugal  force 
alone.  In  completing  the  final  computations  the  results  may  be 
modified  to  allow  for  these.  In 
the  diagrams  here  shown  their 
effects  have  been  very  much  ex- 
aggerated for  clearness  in  the 
cuts. 

191.  Governors    with    Hori- 
zontal   Spindle.  —  Spring   gover- 
nors are  powerful,  as  the  complete 
computations  in  the  next   case 
will  show,  and  are  therefore  well 
adapted  to  cases  where  the  move- 

ment of  the  valve  gear  is  difficult  and  unsteady. 

When  such  a  governor  is  placed  with  horizontal  spindle  such 
as  Fig.  135  the  effects  of  the  weights  are  balanced  and  the  spring 
alone  balances  the  centrifugal  force. 

192.  Belliss  and  Morcom  Governor.  —  One  other  governor  of 
this  general  type  may  be  discussed  in  concluding  this  section. 
It  is  a  form  of  governor  now  much  in  use  and  the  one  shown  in 
the  illustration,  Figs.  136  and  137,  is  used  by  Belliss  and  Morcom 
of  Birmingham,  England,  in  connection  with  their  high-speed 
engine.     The  governor  is  attached  to  the  crankshaft,  and  therefore 
the  weights  revolve  in  a  vertical  plane,  so  that  their  gravity  effect 
is  zero.     There  are  two  revolving  weights  W  with  their  centers 
of  gravity  at  G  and  these  are  pivoted  to  the  spindle  by  pins  A. 


224 


THE  THEORY  OF  MACHINES 


FIG.  135. — Governor  with  gravity  effect  neutralized. 


FIG.  136. — Belliss  and  Morcom  governor. 


GOVERNORS 


225 


w 


Between  the  weights  there  are  two  springs  S  fastened  to  the 
former  by  means  of  pins  at  B.  The  balls  operate  the  collar  C, 
which  slides  along  the  spindle,  thus  operating  the  bell-crank  lever 
DFV,  which  is  pivoted  to  the  engine  frame  at  F  and  connected 
at  V,  by  means  of  a  vertical  rod,  to  the  throttle  valve  of  the 
engine.  There  is  an  additional  compensating  spring  Sc  with  its 
right-hand  end  attached  to  the  frame  and  its  left-hand  end  con- 
nected to  the  bell-crank  lever  DFV  at  Ht  there  being  a  hand 
wheel  at  this  connection  so  that  the  tension  in  the  spring  may  be 
changed  within  certain  limits  and  thus  the  engine  speed  may  be 
varied  to  some  extent.  This  spring  will  easily  allow  the  operator 
to  run  the  engine  at  nearly 
5  per  cent,  above  or  below 
normal. 

The  diagrammatic  sketch  of 
the  governor,  shown  in  Fig. 
137,  enables  the  different  parts 
to  be  distinctly  seen  as  well  as 
the  eolations  of  the  various 
points.  It  will  be  noticed  that 
this  governor  differs  from  all 
the  others  already  described 
in  that  part  of  the  centrifugal 
force  is  directly  taken  up  by 
the  springs  S,  while  the  forces 
acting  on  the  sleeve  are  due  to 
the  dead  weight  of  the  valve 
V  and  its  rod,  and  the  slightly 
unbalanced  steam  pressure 
(for  the  valve  is  nearly 
balanced  against  steam  pres- 
sure) on  the  valve,  and  in  addition  to  these  forces  there  is  the 
pressure  due  to  the  spring  Se.  The  governor  is  very  efficiently 
oiled  and  it  is  found  by  actual  experiment  that  the  frictional  effect 
may  be  practically  neglected. 

In  this  case  it  will  be  advisable  to  draw  the  moment  curve 
for  the  governor  as  well  as  the  C  curve  and  from  the  latter  the 
usual  information  may  be  obtained.  As  this  moment  curve 
presents  no  difficulties  it  seems  unnecessary  to  put  the  investi- 
gation in  a  mathematical  form  as  the  formulas  become  lengthy 
on  account  of  the  disposition  of  the  parts.  An  actual  case  has 

15 


w 


FIG.    137. — Belliss  and  Morcom 
governor. 


226 


THE  THEORY  OF  MACHINES 


been  worked  out  and  the  results  are  given  herewith  and  show  the 
effects  of  the  various  parts  of  the  governor. 

193.  Numerical  Example. — The  governor  here  selected,  is 
attached  to  the  crankshaft  of  an  engine  which  has  a  normal  mean 
speed  of  525  revolutions  per  minute  although  the  actual  speed 
depends  upon  the  load  and  the  adjustment  of  the  spring  Sc. 
The  governor  spindle  also  rotates  at  the  same  speed  as  the  engine. 
The  two  springs  S  together  require  a  total  force  of  112  pds.  for 
each  inch  of  extension,  while  the  spring  Sc  requires  220  pds.  per 
inch  of  extension,  the  springs  having  been  found  on  calibration 
to  be  extremely  uniform.  Each  of  the  revolving  masses  has  an 
effective  weight  of  10.516  Ib.  and  the  radius  of  rotation  of  the 
center  of  gravity  varies  from  4.20  in.  to  4.83  in.  The  other 
dimensions  are:  e  =  4  in.  radius,  b  =  3.2  in.,  c  =  3.5  in.,  a  = 
3.56  in.,  d  =  10.31  in.,/  =  4.67  in.,  g  =  5.31  in. 

The  weight  of  the  valve  spindle,  valve  and  parts  together  with 
the  unbalanced  steam  pressure  under  full-load  normal  conditions 
is  20  Ib. 

The  following  table  gives  the  results  for  the  governor  for  four 
different  radii  of  the  center  of  gravity  G,  all  the  moments  being 
expressed  in  inch-pound  units,  when  reduced  to  the  equivalent 
moment  about  the  pivots  A  of  the  balls. 


BELLISS  AND  MOECOM  GOVERNOR 


Speed, 
revolutions 
per 
minute 

Radius 
of  rota- 
tion of  G, 
inches 

Centrifu- 
gal force, 
pounds 

Moment 
about  A, 
inch- 
pounds 

(i) 

Moment 
of  main 
springs 
about  A, 
inch- 
pounds 

(2) 
Moment 
of  spring 
Se  about 
A,  inch- 
pounds 

(3) 
Moment 
due  to 
valve 
weight 
about  A, 
inch- 
pounds 

Sum  of 
(D,    (2) 
and  (3), 
inch- 
pounds 

508 

4.47 

345.5 

1,220 

1,083 

115 

16 

1,214 

526 

4.73 

392.0 

1,364 

1,196 

143 

16 

1,355 

529 

4.78 

400.0 

1,388 

1,223 

145 

16 

1,384 

532 

4.83 

410.0 

1,418 

1,250 

148 

16 

1,414 

In  examining  the  table  it  will  be  observed  that  the  sum  of 
columns  (1),  (2)  and  (3)  is  always  a  little  less  than  the  moment 
due  to  the  centrifugal  force.  As  the  results  are  all  computed 
from  measurements  made  on  the  engine  during  operation,  there 
is  possibly  a  slight  error  in  the  dimensions,  and  further  the  effect 
of  centrifugal  force  on  the  springs  S  will  make  some  difference. 


GOVERNORS 


227 


The  results  agree  very  well,  however,  and  show  that  the  calcula- 
tions agree  with  actual  conditions. 

The  results  are  plotted  in  Fig.  138,  the  left-hand  part  of  the 
curves  being  dotted.  The  reason  of  this  is  that  the  observations 
below  508  revolutions  per  minute  were  taken  when  the 
engine  was  being  controlled  partly  by  the  throttle  valve,  and 
do  not  therefore  show  the  action  of  the  governor  fairly;  the  points 
are,  however,  useful  in  showing  the  tendency  of  the  curves  and 
represent  actual  positions  of  equilibrium  of  the  governor. 

The  effect  of  the  weight  of  the  valve  and  unbalanced  steam 
pressure  are  almost  negligible,  so  that  the  power  of  the  governor 


sating   8pri2i_H£—- 

Valve    Weight 


Radius  r 
3.70     3.8     3.9     4.0      4.1      4.2      4.3      4.4       4.5     4.6      4.7     4.8          Inches 

FIG.  138. 

does  not  need  to  be  large,  but  the  spring  Sc  produces  an  appre- 
ciable effect  amounting  to  about  11  per  cent,  of  the  total  at  the 
highest  speed.  If  the  compensating  spring  Sc  were  removed, 
the  governor  would  run  at  a  lower  speed. 

Joining  any  point  on  the  moment  curve  to  the  origin  0,  as 
has  been  done  on  the  figure,  shows  that  the  governor  is  stable. 

The  sensitiveness  and  powerfulness  may  be  found  from  the 
C  curve  shown  at  Fig.  139.  At  the  radius  4.47  in.  the  centri- 
fugal force  is  345.5  pds.,  and  if  the  origin  be  joined  to  this  point 
and  the  line  produced  it  will  cut  the  radius  4.83  in.  at  373.5  pds., 
whereas  the  actual  C  is  410  pds.  The  sensitiveness  then  is 


228 


THE  THEORY  OF  MACHINES 


o^o'rx  =  0.0465  or  4.65  per  cent.     From  the  speeds 

^(^±1(J  -J-   O/O.OJ 

/coo  508") 

the  corresponding  result  would  be  1x^500  _u  508")  =  0.0442  or  4.42 

per  cent.,  which  agrees  quite  closely  with  the  former  value. 
The  moment  curves  cannot  be  used  directly  for  the  determina- 
tion of  the  power  of  the  governor  because  areas  on  the  diagram 
do  not  represent  work  done.  If  the  power  is  required,  then  the 
base  must  be  altered  either  so  as  to  represent  equal  angles  passed 
through  by  the  ball  arm,  or  more  simply  by  use  of  the  C  curve 
plotted  on  Fig.  139.  It  will  be  seen  that  the  C  curve  differs 


^  Pounds 


(00 


300 


200 


100 


3.7      3.8     3.9     4.0     4.1 


4.2     4.3       4.4      4.5 

FIG.  139. 


4.6       4.7      4.8       Inches 


very  little  in  character  from  the  moment  curve.     The  power 
of  the  governor  is  only  about  11.6  ft.-pds. 

The  computations  on  this  one  governor  will  give  a  good  general 
idea  of  the  relative  effects  of  the  different  parts  in  this  style  of 
governor,  and  also  show  that  spring  governors  of  this  class 
possess  some  advantages. 


THE   INERTIA   GOVERNOR,   FREQUENTLY   CALLED    THE  SHAFT 

GOVERNOR 

194.  Reasons  for  Using  this  Type. — The  shaft  governor  was 
probably  originally  so  named  because  it  is  usually  secured  to  the 
crankshaft  of  an  engine  and  runs  therefore  at  the  engine  speed. 
In  recent  practice,  however,  certain  spring  governors,  such  as  the 
Belliss  and  Morcom  governor,  are  attached  to  the  crankshaft 


GOVERNORS  229 

and  yet  these  scarcely  come  under  the  name  of  shaft  governors. 
The  term  is  more  usually  restricted  to  a  governor  in  which  the 
controlling  forces  differ  to  some  extent  from  those  already  dis- 
cussed. This  type  of  governor  is  not  nearly  so  old  as  the  others 
and  was  introduced  into  America  mainly  as  an  adjunct  to  the 
high-speed  engine. 

On  this  continent  builders  of  high  (rotative) -speed  engines 
have  almost  entirely  governed  them  by  the  method  first  men- 
tioned at  the  beginning  of  this  chapter,  that  is  by  varying  the 
point  of  cut-off  of  the  steam,  and  in  order  to  do  this  they  have 
usually  changed  the  angle  of  advance  and  also  the  throw  of 
the  eccentric  by  means  of  a  governor  which  caused  the  center 
of  the  eccentric  to  vary  in  position  relative  to  the  crank  according 
to  the  load,  the  result  will  be  a  change  in  all  the  events  of  the 
stroke.  The  eccentric's  position  is  usually  directly  controlled  by 
the  governor,  and  hence  it  is  necessary  to  have  a  powerful  gover- 
nor or  else  the  force  required  to  move  the  valve  may  cause  very 
serious  disturbances  of  the  governor  and  render  it  useless.  Again 
as  the  governor  works  directly  on  the  eccentric,  it  is  convenient 
to  have  it  on  the  crankshaft. 

Governors  of  this  class  also  possess  another  peculiarity  In- 
those  already  described  the  pins  about  which  the  balls  swung 
were  in  all  cases  perpendicular  to  the  axis  of  rotation,  so  that  the- 
balls  moved  out  and  in  a  plane  passing  through  this  axis.  In 
the  shaft  governor,  on  the  other  hand,  the  axis  of  the  pins  is 
parallel  with  the  axis  of  rotation  and  the  weights  move  out  and 
in  in  the  plane  in  which  they  rotate.  While  this  may  at  first 
appear  to  be  a  small  matter,  it  is  really  the  point  which  makes  this 
class  of  governor  distinct  from  the  others  and  which  brings  into 
play  inertia  forces  during  adjustment  that  are  absent  in  the 
other  types.  Such  governors  may  be  made  to  adjust  themselves 
to  their  new  positions  very  rapidly  and  are  thus  very  valuable 
on  machinery  subjected  to  sudden  and  frequent  changes  of  load. 

195.  Description. — One  make  of  shaft  governor  is  shown  at 
Fig.  140,  being  made  by  the  Robb  Engineering  Co.,  Amherst, 
Nova  Scotia,  and  is  similar  to  the  Sweet  governor.  In  this  make 
there  is  only  one  rotating  weight  W,  the  centrifugal  effect  of 
which  is  partly  counteracted  by  the  flat  leaf  spring  S,  to 
which  the  ball  is  directly  attached.  The  eccentric  E  is  pivoted 
by  the  pin  P  to  the  flywheel,  and  an  extension  of  the  eccen- 
tric is  attached  by  the  link  b  to  the  ball  W.  The  wheel 


230 


THE  THEORY  OF  MACHINES 


rotating  in  the  sense  shown,  causes  the  ball  to  try  to 
move  out  on  a  radial  line,  which  movement  is  resisted  by  the 
spring  S.  As  the  ball  moves  out,  due  to  increased  speed,  the 
eccentric  sheave  swings  about  P,  and  thus  the  center  of  the  eccen- 
tric will  take  up  a  position  depending  upon  the  speed.  Two  stops 
are  provided  to  limit  the  extreme  movement  of  the  eccentric 
and  ball. 

Other  forms  of  governor  are  shown  later  at  Fig.  143  and  at 
Fig.  147,  these  having  somewhat  different  dispositions  of  the 
parts. 


FIG.   140. — Robb  inertia  governor. 

Powerfulness  in  such  governors  is  obtained  by  the  use  of 
heavy  weights  moving  at  high  speed,  for  example  in  one  governor 
the  revolving  weight  is  80  Ib.  and  it  revolves  in  a  circle  of  over 
29  in.  radius  at  200  revolutions  per  minute,  dimensions  which 
should  be  compared  with  those  in  the  governors  already  discussed. 

196.  Conditions  to  be  Fulfilled. — The  conditions  to  be  ful- 
filled are  quite  similar  to  those  in  other  spring  governors  so  that 
only  a  brief  discussion  will  be  necessary,  which  may  be  illustrated 
in  the  following  example. 

Let  A,  Fig.  141,  represent  a  disc  rotating  about  a  center  0 
at  n  revolutions  per  minute,  and  let  this  disc  have  a  weight  w 


GOVERNORS 


231 


mounted  on  it  so  that  it  may  move  in  and  out  along  a  radial  line 
as  indicated,  and  further  let  the  motion  radially  be  resisted  by  a 
spring  S  which  is  pivoted  to  the  disc  at  E.  Let  the  spring  pull 
per  foot  of  extension  be  $  pds.,  and  let  the  weight  be  in  equilibrium 
at  distance  r  ft.  from  0,  the  extension  of  the  spring  at  this  in- 


w 


stant  being  a  ft.     The  centrifugal  force  on  the  ball  is  C  =  —  rco2 

& 

pds.  where  co   is  the  angular  velocity  of  the  disc  in  radians  per 

2m 


second,  and  since  co 


-AQ-,  therefore 


g 


=  0.000341 


where  r  is  in  feet.     For  the  same    position  the   spring  pull  will 
be  Sa  pds.,  so  that  for  equilib- 
rium Sa  =  C  or 


0.000341  wrn*, 


that  is, 


S  =  0.000341  w-n2. 
a 

To  make  the  meaning  of 
this  clear  it  will  be  well  to 
take  a  numerical  example, 
and  let  it  be  assumed  that  the 
weight  w  =  25  Ib.  and  the 
speed  is  200  revolutions  per 
minute.  Three  cases  may  be 

considered,  according  to  whether  r  is  equal  to,  greater  than  or 
less  than  a,  and  these  will  result  as  follows: 

1.  r  =  a  =  1ft.,  S  =  0.000341  X  25  X  j  X  2002  =  341  pds. 

2.  r  =  lft.,  a  =  0.57  ft.,  S  =  0.000341  X25X^X  2002  =  600  pds. 

3.  r  =  1  ft.,  a  =  1.19  ft.,  S  =  0.000341  X  25  X  j^X  2002  =  288 

pds. 

So  that,  as  the  formula  shows,  the  spring  strength  depends 
upon  the  relation  of  r  to  a. 

The  resulting  conditions  when  the  ball  is  10  in.,  12  in.  and  14 
in.  respectively  from  the  center  of  rotation  with  the  three  springs, 
are  set  down  in  the  following  table  and  in  Fig.  142. 


232 


THE  THEORY  OF  MACHINES 


Radius  r,  inches 

Centrifugal 
force  at  200 
rev. 

Spring  pull 

S  =  288 

S  =  341 

S  =  600 

10 

284 

293 

284 

241 

12 

341 

341 

341 

341 

14 

398 

389 

398 

441 

For  the  spring  S  =  288  pds.  per  foot  of  extension  it  is  seen  that 
at  the  smaller  radius  the  spring  pull  is  higher  than  the  centrifugal 
force  or  the  disc  must  run  at  a  higher  speed  than  200  revolutions 
for  equilibrium,  while  at  the  outer  radius  the  spring  pull  is  too  lo\v 
and  the  speed  must  be  below  200  revolutions  for  equilibrium, 


341 


293 


-1.19ft; 


FIG.  142. 


that  is  the  speed  should  decrease  as  the  ball  moves  out.  With 
spring  S  =  600  exactly  the  reverse  is  true,  or  the  speed  must 
increase  as  the  ball  moves  out,  while  for  spring  S  =  341  the  speed 
will  be  constant  for  all  positions  of  the  ball. 

The  spring  S  =  341  is  properly  designed  and  set  for  isochronism, 
but  evidently  there  is  no  force  holding  the  ball  anywhere  and  the 
slightest  push  would  send  it  oscillating  along  the  scale,  that  is, 
it  lacks  stability.  The  spring  S  =  288  also  gives  an  unstable 
arrangement  for  the  reason  that  the  centrifugal  force  increases 
and  decreases  faster  than  the  spring  pull,  and  thus  if  the  ball 
happened  to  be  12  in.  from  the  center  and  was  disturbed  it 
would  instantly  fly  to  the  inner  or  outer  extreme  stop.  However, 


GOVERNORS 


233 


spring  S  =  600  gives  a  stable  arrangement,  because  whenever 
the  ball  is  at  say  12  in.  from  the  center  and  any  force  pushes  it 
away  it  immediately  tries  to  return  to  this  position,  and  will  do 
so  on  account  of  the  preponderating  effect  of  the  spring  force 
acting  upon  it,  unless  there  should  be  a  change  of  speed  forcing 
it  to  the  new  position,  but  to  each  speed  there  is  a  definitely  fixed 
position  of  the  ball.  It  is  to  be  noticed  that  the  curve  for  S  = 
600,  is  always  steeper  than  any  line  from  it  to  the  origin  0,  which 
has  been  already  given  as  a  condition  of  stability  (Sec.  183(2)). 

The  gain  in  stability  is,  however,  made  at  a  sacrifice  in 
sensitiveness.  For  the  spring  S  =  600  the  speed  changes  from 
184  to  211  revolutions  per  minute  or  the  sensitiveness  is 

1/^91     4-  =  0-136   or    1^.6   per   cent.,    while    with    spring 


FIG.  143. — Buckeye  and  McEwen  governors. 

S  =  288  the  range  of  speeds  is  from  198  to  203  revolutions  per 
minute  or  the  sensitiveness  is  2.4  per  cent. 

197.  Analysis  of  the  Governor. — Having  now  discussed  the 
conditions  of  stability  and  isochronism  and  the  effect  the  design 
of  the  spring  has  on  them,  a  complete  analysis  of  the  governor 
may  be  made. 

Two  forms  of  governor  are  shown  in  Fig.  143  and  these  show  a 
somewhat  different  disposition  of  the  revolving  weights.  The 
one  on  the  left  is  used  by  the  Buckeye  Engine  Co.  and  has  two 
revolving  weights  W  connected  by  arms  b  to  the  pivots  P.  The 
centrifugal  force  is  resisted  by  springs  S  attached  to  b  and  to  the 
flywheel  rim  at  K.  The  ends  of  the  links  b  are  connected  at  H 
to  links  attached  to  the  eccentric  E  at  C  and  the  operation  of 


234 


THE  THEORY  OF  MACHINES 


the  weights  revolves  E  and  changes  the  steam  distribution. 
Auxiliary  springs  D  oppose  springs  S  at  inner  positions  of  weights 
W. 

The  right-hand  figure  shows  the  McEwen  governor  having 
two  unequal  weights  W\  and  Wz  cast  on  a  single  bar,  the  com- 
bined center  of  gravity  being  at  G,  and  the  pivot  connection  to  the 
wheel  is  P.  There  is  a  single  spring  S  attached  to  the  weights  at 
H  and  to  the  wheel  at  K.  There  is  a  dashpot  at  D  attached  to 
the  wheel  and  to  the  weight  at  J5;  this  consists  of  a  cylinder  and 
piston,  the  latter  being  prevented  from  moving  rapidly  in  the 
cylinder.  The  purpose  of  the  dashpot  is  to  prevent  oscillations 


of  the  weight  during  adjustment  and  to  keep  it  steady,  but  after 
adjustment  has  been  made  D  has  no  effect  on  the  conditions  of 
equilibrium.  In  this  governor  a  frictionless  pin  is  provided  at 
P  by  the  use  of  a  roller  bearing.  The  valve  rod  is  at  E. 

A  diagrammatic  drawing  of  these  two  governors,  which  may 
be  looked  upon  as  fairly  representative  of  this  class,  is  given  at 
Fig.  144,  similar  letters  being  used  in  both  cases. 

Let  the  wheel  revolve  about  A,  Fig.  143,  with  angular  velocity 
w  radians  per  second  and  let  F  denote  the  spring  pull  when  the 
center  of  gravity  G  is  at  radius  r  from  A;  further,  let  di  and  dz 
in.  represent  the  shortest  distances  from  the  weight  pivot  P 
to  the  directions  of  r  and  S  respectively.  Then  for  equilibrium 


GOVERNORS  235 

the  moments  about  P  due  to  the  centrifugal  force  C  and  to  the 
spring  pull  S  must  be  equal  if,  for  the  present,  the  effect  of 
gravity  and  of  the  forces  required  to  move  the  valve  are  neglected. 
That  is: 

Cdi  =  Sdz  in.-pds. 

w  0     . 

or  —  rco2di  =  Sdz  m.-pds. 

In  such  an  arrangement  as  shown  the  effect  of  the  forces  re- 
quired to  move  the  valves  is  frequently  quite  appreciable  and 
is  generally  also  variable,  as  is  also  the  effect  of  friction  and 
gravity,  although  usually  gravity  is  relatively  so  small  that  it 
may  be  neglected.  If  it  is  desired  to  take  these  into  account 
then 


i  =  Sd2  +  moment  due  friction,  valve  motion  and  gravity. 

Denote  the  distance  AP  by  a  and  the  shortest  distance  from 
G  to  AP  by  x;  thus  a  is  constant  but  x  depends  on  the  position 
of  the  balls.  From  similar  triangles  it  is  evident  that  rd\  =  ax 
and  therefore 

w  w    . 


Thus  the  moment  due  to  the  centrifugal  force  is,  for  a  given 
speed,  variable  only  with  x  and  hence  the  characteristics  of  the 
governor  are  very  well  shown  on  a  curve1  in  which  the  base  repre- 
sents values  of  x  and  vertical  distances  the  centrifugal  moments 

—  uzax. 
9 

Such  a  curve  is  shown  below,  Fig.  144,  and  the  shape  of  the 
curve  here  represents  a  stable  governor  since  it  is  steeper  at  all 
points  than  the  line  joining  it  to  the  origin  0.  From  this  curve 
information  may  be  had  as  to  stability  and  sensitiveness,  but  the 
power  of  the  governor  cannot  be  determined  without  either 
placing  the  curve  on  a  base  which  represents  the  angular  swing 
of  the  balls  about  P  or  else  by  obtaining  a  C  curve  on  an  r  base 
as  in  former  cases. 

If  co  is  constant,  or  the  governor  is  isochronous,  M  varies 
directly  with  x  or  the  moment  curve  is  a  straight  line  passing 
through  the  origin  0. 

Having  obtained  the  M  curve  in  this  way  the  moment  curves 

1  For  more  complete  discussion  of  this  method  see  TOLLE,  "  Die  Regelung 
der  Kraftmaschinen." 


236 


THE  THEORY  OF  MACHINES 


about  P  corresponding  to  gravity,  friction  of  the  valves  and  parts 
and  also  those  necessary  to  operate  the  valves  are  next  found, 
these  three  curves  also  being  plotted  on  the  x  base,  and  the 
difference  between  the  sum  of  these  three  moments  and  the  total 
centrifugal  moment  will  give  the  moment  which  must  be  pro- 
vided by  the  spring  which  is  Sd2  in.-pds.  From  the  curve  giving 
Sd2  the  force  S  may  be  computed  by  dividing  by  <i2  and  these 
values  of  S  are  most  conveniently  plotted  on  a  base  of  spring 
lengths,  from  which  all  information  for  the  design  of  the  spring 
may  readily  be  obtained  (see  Sec.  190). 

In  order  that  the  relative  values  of  the  different  quantities 
may  be  understood,  Fig.  145  shows  these  curves  for  a  Buckeye 


Eccentric  Friction 

FIG.  145. 


Valve  Friction 


governor,  in  which  the  gravity  effect  is  balanced  by  using  two 
revolving  weights  symmetrically  located.  Friction  of  the  valve 
and  eccentric  and  the  moment  required  to  move  the  valve  are 
all  shown  and  the  curves  show  how  closely  the  spring-moment 
and  centrifugal-moment  curves  lie  together.  The  curves  are 
drawn  from  the  table  given  by  Trinks  and  Housom,  in  whose1 
treatise  all  the  details  of  computing  the  results  is  shown  so  as  to 
be  clearly  understood.  The  governor  has  a  powerf ulness  of  nearly 
600  ft.-pds. 

RAPIDITY  OF  ADJUSTMENT 

198.  The  inertia  or  shaft  governor  is  particularly  well  adapted 
to  rapid  adjustment  to  new  conditions  and  it  is  often  made  so 
1  THINKS  and  HOUSOM,  "Shaft  Governors." 


GOVERNORS 


237 


that  it  will  move  through  its  entire  range  in  one  revolution,  which 
often  means  only  a  small  fraction  of  a  second.  The  rapidity  of 
this  adjustment  depends  almost  entirely  upon  the  distribution  of 
the  revolving  weight  and  not  nearly  so  much  upon  its  magnitude. 
For  a  given  position  of  the  parts  the  only  force  acting  is  centrif- 
ugal force  already  discussed  but  during  change  of  position  the 
parts  are  being  accelerated  and  forces  due  to  this  also  come  into 
play.  Fig.  146  represents  seven  different  arrangements  of  the 
weights;  in  five  of  these  the  weight  is  concentrated  into  a  ball 
with  center  of  gravity  at  G  and  hence  with  very  small  moment 
of  inertia  about  G,  so  that  the  torque  required  to  revolve  such 
a  weight  at  any  moderate  acceleration  will  not  be  great;  the 
opposite  is  true  of  the  two  remaining  cases,  however,  the  weight 


M 


being  much  elongated  and  having  a  large  moment  of  inertia 
about  G. 

Assuming  a  sudden  increase  of  speed  in  all  cases,  then  at  (a) 
this  only  increases  the  pressure  on  the  pin  B  because  BG  is 
normal  to  the  radius  AG,  at  (6)  an  increase  in  speed  will  produce 
a  relatively  large  turning  moment  about  the  pin  which  is  shown 
at  A.  Comparing  (c)  and  (d)  with  (a)  and  (b)  it  is  seen  that  the 
torque  in  the  former  cases  is  increased  at  (c)  because  in  addition 
to  the  acceleration  of  G  there  is  also  an  angular  acceleration  about 
G,  whereas  at  (d)  G  is  stationary  and  yet  there  is  a  decided  torque 
due  to  its  angular  acceleration.  At  (e),  (/)  and  (</)  the  sense  of 
rotation  is  important  and  if  an  increase  in  speed  occur  in  the 
first  and  last  cases  the  accelerating  forces  assist  in  moving  the 


238  THE  THEORY  OF  MACHINES 

weights  out  rapidly  to  their  new  positions,  whereas  at  (/)  the 
accelerating  forces  oppose  the  movement. 

Space  prevents  further  discussion  of  this  matter  here,  but  it 
will  appear  that  the  accelerating  forces  may  be  adjusted  in  any 
desired  way  to  produce  rapid  changes  of  position,  the  weights 
being  first  determined  from  principles  already  stated  and  the 
distribution  of  these  depending  on  the  inertia  effects  desired. 
Chapter  XV  will  assist  the  reader  in  understanding  these  forces 
more  definitely. 


FIG.   147. — Rites  governor. 

A  form  of  governor  made  by  Rites,  in  which  the  inertia  forces 
play  a  prominent  part  during  adjustment  is  shown  at  Fig.  147. 
The  revolving  weights  are  heavy  and  are  set  far  apart,  but  their 
center  of  gravity  G  is  fairly  close  to  A  so  that  the  centrifugal 
moment  is  relatively  small.  In  a  governor  for  a  10  by  10-in. 
engine  the  weight  W  was  over  120  Ib.  and  the  two  weight  centers 
were  32  in.  apart. 

QUESTIONS  ON  CHAPTER  XII 

1.  Define  a  governor.  What  is  the  difference  between  the  functions  of  a 
governor  and  a  flywheel? 


GOVERNORS  239 

2.  What  is  the  height  of  a  simple  governor  running  at  95  revolutions  per 
minute  ? 

3.  What  is  meant  by  an  isochronous  governor?     Is  such  a  governor  desir- 
able or  not?     Why? 

4.  Explain  fully  the  terms  stability  and  powerfulness. 

5.  Prove  that  in  a  governor  where  the  balls  move  in  a  paraboloid  of  revo- 
lution, h  is  constant  and  the  governor  is  isochronous. 

6.  What  are  the  advantages  of  the  Porter  governor? 

7.  Using  the  data,  n\  =  100,  n2  =  110,  prove  that  -r  =    —  2  — 

fl  CO 

^  8.  Compare  the  sensitiveness  of  a  simple  and  a  Porter  governor  at  115 
revolutions  per  minute  and  with  a  sleeve  travel  of  %  in.,  taking  W  =  120  Ib. 
and  w  =  15  Ib. 

9.  Analyze  the  following  governor  for  sensitiveness  and  power  (see  Fig. 
129): 

n  =  130,  W  =  110,  w  =  12,  h  =  12K,   BM  =  3^,  h  =  1(%  «i  =  0, 
«z  =  2^,  sleeve  travel  2^  in. 

10.  Design  a  Porter  governor  for  a  speed  of  170  revolutions  per  minute 
with  a  speed  variation  of  5  percent,  each  way,  travel  2^  in.,  power  35  ft.-pds. 

11.  In  a  governor  of  the  type  of  Fig.  133,  a  =  2. 1  in.;  b  =  0.75  in.,  dis- 
tance between  pivots  2%  in.  inner  radius  of  ball  1.6  in.,  weight  per  ball  1% 
Ib.,  travel  Y±  in.  and  speed  250.     Design  the  spring  for  5  per  cent,  variation. 

12.  What  are  the  advantages  of  the  shaft  governor?     Show  how  the  dis- 
tribution of  the  weight  affects  the  rapidity  of  adjustment. 


CHAPTER  XIII 
SPEED  FLUCTUATIONS  IN  MACHINERY 

199.  Nature  of  the  Problem. — The  preceding  chapter  deals 
with  governors  which  are  used  to  prevent  undue  variations  in 
speed  of  various  classes  of  machinery,  the  governor  usually  con- 
trolling the  supply  of  energy  to  the  machine  in  a  way  to  suit  the 
work  to  be  done  and  so  as  to  keep  the  mean  speed  of  the  machine 
constant.     The  present  chapter  does  not  deal  with  this  kind  of  a 
problem  at  all,  but  in  the  discussion  herein,  it  is  assumed  that  the 
mean  speed  of  the  machine  is  constant  and  that  it  is  so  controlled 
by  a  governor  or  other  device  as  to  remain  so. 

Iri  addition  to  the  variations  in  the  mean  speed  there  are 
variations  taking  place  during  the  cycle  of  the  machine  and  which 
may  cause  just  as  much  trouble  as  the  other.  Everyone  is  famil- 
iar with  the  small  direct-acting  pump,  and  knows  that  although 
such  a  pump  may  make  80  strokes  per  minute,  for  example, 
and  keep  this  up  with  considerable  regularity,  yet  the  piston 
moves  very  much  faster  at  certain  times  than  others,  and  in  fact 
this  variation  is  so  great  that  larger  pumps  are  not  constructed  in 
such  a  simple  way.  With  the  larger  pumps,  on  which  a  crank 
and  flywheel  are  used,  an  observer  frequently  notices  that, 
although  the  mean  speed  is  perfectly  constant,  yet  the  flywheel 
speed  during  the  revolution  is  very  variable.  Where  a  steam 
engine  drives  an  air  compressor,  these  variations  are  usually 
visible,  at  certain  parts  of  the  revolution  the  crankshaft  almost 
coming  to  rest  at  times.  These  illustrations  need  not  be  multi- 
plied, but  those  quoted  will  suffice.  The  speed  variations  which 
occur  in  this  way  during  the  cycle  are  dealt  with  in  this  chapter. 

200.  Cause  of  Speed  Fluctuations. — The  flywheel  of  an  engine 
or  punch  or  other  similar  machine  is  used  to  store  energy  and  to 
restore  it  to  the  machine  according  to  the  demands.     Consider, 
for  example,  the  steam  engine ;  there  the  energy  supplied  by  the 
steam  at  different  parts  of  the  stroke  is  not  constant,  but  varies 
from  time  to  time;  at  the  dead  centers  the  piston  is  stationary  and 
hence  no  energy  is  delivered  by  the  working  fluid,  whereas  when 

240 


SPEED  FLUCTUATIONS  IN  MACHINERY        241 

the  piston  has  covered  about  one-third  of  its  stroke,  energy  is  being 
delivered  by  the  steam  to  the  piston  at  about  its  maximum  rate, 
since  the  piston  is  moving  at  nearly  its  maximum  speed  and 
the  steam  pressure  is  also  high,  as  cutoff  has  not  usually 
taken  place.  Toward  the  end  of  the  stroke  the  rate  of  delivery 
of  the  energy  by  the  steam  is  small  because  the-  steam  pressure 
is  low  on  account  of  expansion  and  the  piston  is  moving  at  slow 
speed.  During  the  return  stroke  the  piston  must  supply  energy 
to  the  steam  in  order  to  drive  the  latter  out  of  the  cylinder. 

Now  the  engine  above  referred  to  may  be  used  to  drive  a  pump 
or  an  air  compressor  or  a  generator  or  any  other  desired  machine, 
but  in  order  to  illustrate  the  present  matter  it  will  be  assumed  to 
be  connected  to  a  turbine  pump,  since,  in  such  a  case,  the  pump 
offers  a  constant  resisting  torque  on  the  crankshaft  of  the  engine. 
The  rate  of  delivering  energy  by  the  working  fluid  is  variable,  as 
has  already  been  explained  ;  at  the  beginning  of  the  revolution  it 


Jerque  Required 
V     by  Load 


O  Mf  N'      ^— '  R'  S'        ^A; 

FIG.  148. 

is  much  less  than  that  required  to  drive  the  pump,  a  little  further 
on  it  is  much  greater  than  that  required,  while  further  on  again 
the  steam  has  a  deficiency  of  energy,  and  so  on. 

At  this  point  it  will  be  well  to  refer  again  to  Fig.  101  which 
has  been  reproduced  in  a  modified  form  at  Fig.  148  and  shows  in 
a  very  direct  and  clear  way  these  important  features.  During 
the  first  part  of  the  outstroke  it  is  evident  that  the  crank  effort 
due  to  the  steam  pressure  is  less  than  that  necessary  to  drive 
the  load;  this  being  the  case  until  M  is  reached,  at  which  point 
the  effort  due  to  the  steam  pressure  is  just  equal  to  that  necessary 
to  drive  the  load;  thus  during  the  part  OMf  of  the  revolution  the 
input  to  the  engine  being  less  than  the  output  the  energy  of  the 
links  themselves  must  be  drawn  upon  and  must  supply  the  work 
represented  by  OML.  But  the  energy  which  may  be  obtained 
from  the  links  will  depend  upon  the  mass  and  velocity  of  them, 
the  energy  being  greater  the  larger  the  mass  and  the  greater  the 
velocity,  the  result  is  that  if  the  energy  of  the  links  is  decreased 

16 


242  THE  THEORY  OF  MACHINES 

by  drawing  from  them  for  any  purpose,  then  since  the  mass  of 
the  links  is  fixed  by  construction,  the  only  other  thing  which 
may  happen  is  that  the  speed  of  the  links  must  decrease. 

In  engines  the  greater  part  of  the  weight  in  the  moving  parts  is 
in  the  flywheel  and  hence,  from  what  has  been  already  said  if 
energy  is  drawn  from  the  links  then  the  velocity  of  the  flywheel 
will  decrease  and  it  will  continue  to  decrease  so  long  as  energy  is 
drawn  from  it.  Thus  during  OMr  the  speed  of  the  flywheel  will 
fall  continually  but  at  a  decreasing  rate  as  M '  is  approached,  and 
at  this  point  the  wheel  will  have  reached  its  minimum  speed. 
Having  passed  M'  the  energy  supplied  by  the  steam  is  greater 
than  that  necessary  to  do  the  external  work,  and  hence  there  is  a 
balance  left  for  the  purpose  of  adding  energy  to  the  parts  and 
speeding  up  the  flywheel  and  other  links,  the  energy  available 
for  this  purpose  in  any  position  being  that  due  to  the  height  of 
the  torque  curve  above  the  load  line.  In  this  way  the  speed  of 
the  parts  will  increase  between  M'  and  N'  reaching  a  maximum 
for  this  period  at  N'. 

From  N'  to  Rf  the  speed  will  again  decrease,  first  rapidly  then 
more  slowly,  reaching  a  minimum  again  at  R'  and  from  R  to  Sf, 
there  is  increasing  speed  with  a  maximum  at  S'.  The  flywheel 
and  other  parts  will,  under  these  conditions,  be  continually 
changing  their  speeds  from  minimum  to  maximum  and  vice  versa, 
producing  much  unsteadiness  in  the  motion  during  the  revolu- 
tion. The  magnitude  of  the  unsteadiness  will  evidently  depend 
upon  the  fluctuation  in  the  crank-effort  curve,  if  the  latter  curve 
has  large  variations  then  the  unsteadiness  will  be  increased;  it 
will  also  depend  on  the  weights  of  the  parts. 

In  the  case  of  the  punch  the  conditions  are  the  reverse  of  the 
engine,  for  the  rate  of  energy  supplied  by  the  belt  is.  nearly  con- 
stant but  that  given  out  is  variable.  While  the  punch  runs  light, 
no  energy  is  given  out  (neglecting  friction),  but  when  a  hole  is 
being  punched  the  energy  supplied  by  the  belt  is  not  sufficient, 
and  the  flywheel  is  drawn  upon,  with  a  corresponding  decrease 
in  its  speed,  to  supply  the  extra  energy,  and  then  after  the  hole 
is  punched,  the  belt  gradually  speeds  the  wheel  up  to  normal 
again,  after  which  another  hole  may  be  punched.  To  store  up 
energy  for  such  a  purpose  the  flywheel  has  a  large  heavy  rim 
running  at  high  speed. 

It  will  thus  be  noticed  that  a  flywheel,  or  other  part  serving 
the  same  purpose,  is  required  if  the  supply  of  energy  to  the  ma- 


SPEED  FLUCTUATIONS  IN  MACHINERY        243 

chine,  or  the  delivery  of  energy  by  the  machine,  i.e.,  the  load,  is 
variable;  thus  a  flywheel  is  required  on  an  engine  driving  a  dyna- 
mo or  a  reciprocating  pump,  or  a  compressor,  or  a  turbine  pump; 
also  a  flywheel  is  necessary  on  a  punch  or  on  a  sheet  metal  press. 
It  is  not,  however,  in  general  necessary  to  have  a  flywheel  on  a 
steam  turbo-generator,  or  on  a  motor-driven  turbine  pumping 
set,  or  on  a  water  turbine-driven  generator  set  working  with 
constant  load,  because  in  such  cases  the  energy  supplied  is  always 
equal  to  that  given  out. 

The  present  investigation  is  for  the  purpose  of  determining 
the  variations  or  fluctuations  in  speed  that  may  occur  in  a  given 
machine,  when  the  methods  of  supplying  the  energy  and  also  of 
loading  are  known.  Thus,  in  an  engine-driven  compressor,  hoth 
the  steam-  and  air-indicator  diagrams  are  assumed  known,  as 
well  as  the  dimensions  and  weights  of  the  moving  parts. 

THE  KINETIC  ENERGY  OF  MACHINES 

201.  Kinetic  Energy  of  Bodies. — In  order  to  determine  the 
speed  fluctuations  in  a  machine  it  is  necessary,  first  of  all,  to 
find  the  kinetic  energy  of  the  machine  itself  in  any  given  posi- 
tion and  this  will  now  be  determined. 

If  any  body  has  plane  motion  at  any  instant,  this  motion 
may  be  divided  into  two  parts: 

(a)  A  motion  of  translation  of  the  body. 

(6)  A  motion  of  rotation  of  the  body  about  its  center  of  grav- 
ity. Let  the  weight  of  the  body  be  w  lb.,  then  its  mass  will 

be  m  =  — ,  where  g  is  the  attraction  due  to  gravity  and  is  equal 

J/ 

to  32.16  in  pound,  foot  and  second  units,  and  let  the  body  be 
moving  in  a  plane,  the  velocity  of  its  center  of  gravity  at  the 
instant  being  v  ft.  per  second.  Further,  let  the  body  be  turning 
at  the  same  instant  at  the  rate  of  co  radians  per  second,  and  assume 
that  the  moment  of  inertia  of  the  body  about  its  center  of  gravity 
is  7,  the  corresponding  radius  of  gyration  being  k  ft.,  so  that 
I  =  mk\ 

Then  it  is  shown  in  books  on  mechanics  that  the  kinetic  energy 
of  the  body  is  E  =  %mv2  +  ]4  /co2  =  ^mv2  +  i^ra/<;2a>2  ft.-pds., 
and,  hence,  in  order  to  find  the  kinetic  energy  of  the  body  it  is  neces- 
sary to  know  its  weight  and  the  distribution  of  the  latter  because 
of  its  effect  on  k,  and  in  addition  the  velocity  of  the  center  of 


244  THE  THEORY  OF  MACHINES 

gravity  of  the  weight  and   also   the   angular   velocity   of   the 
body. 

202.  Application  to  Machines. — Let  Fig.  149  represent  a 
mechanism  with  four  links  connected  by  four  turning  pairs,  the 
links  being  a,  b,  c  and  d,  of  which  the  latter  is  fixed,  and  let  Ia, 
Ib  and  Ic  represent  the  moments  of  inertia  of  a,  b  and  c  respective- 
ly about  their  centers  of  gravity,  the  masses  of  the  links  being 
ma,  nib  and  mc.  Assuming  that  in  this  position  the  angular 


FIG.  149. 


velocity  o>  of  the  link  a  is  known,  it  is  required  to  find  the  corre- 
sponding kinetic  energy  of  the  machine. 

Find  the  images  of  P,  Q,  a,  b,  c,  d  and  of  G,  H  and  N,  the  centers 
of  gravity  of  a,  b  and  c  respectively,  by  means  of  the  phorograph 
discussed  in  Chapter  IV.  Now  if  VG,  VH  and  VN  be  used  to  rep- 
resent the  linear  velocities  of  G,  H  and  N  and  also  if  co&  and  o>c  be 
used  to  denote  the  angular  velocities  of  the  links  b  and  c,  it  is 
at  once  known,  from  the  phorograph  (Sees.  66  and  68),  that: 
VQ  =  OG'.u]  VH  =  OH'.u  and  VN  =  ON'.u  ft.  per  second,  and 

61  c' 

fc>6  =  "iT'w  and  coc  =  — co  radians  per  second,  so  that  all  the  neces- 
sary linear  and  angular  velocities  are  found  from  the  drawing. 

203.  Reduced  Inertia  of  the  Machine. — The  investigation  wil) 
be  confined  to  the  determination  of  the  kinetic  energy  of  the 
link  b,  which  will  be  designated  by  Eb,  and  having  found  this 
quantity  the  energy  of  the  other  links  may  be  found  by  a  similar 


SPEED  FLUCTUATIONS  IN  MACHINERY        245 

process.     Since  for  any  body  the  kinetic  energy  at  any  instant 
is  given  by  the  formula: 

Eb  =  y%  mv2  +  M  I"*  ft.-pds. 
Therefore,  Eb  =  ^mb.v2H  +  M  ^bz  ft.-pds. 

'    v  '       2 

Now,          JW  =  mbkbzwb2  =  mbkb2 


Following  the  notation  already  adopted,  it  will  be  convenient 
'to  write  k'b  for  ^rkb,  since  the  length  j-kb  is  the  length  of  the  image 

of  kb  on  the  phorograph.  The  magnitude  of  k'b  is  found  by  draw- 
ing a  line  HT  in  any^  direction  from  H  to  represent  kb  and  find- 
ing T'  by  drawing  H'T'  parallel  to  HT  to  meet  TP  produced  in 
T'  as  indicated  in  Fig.  149;  then  H'T'  is  the  corresponding  value 
of  fc'6. 

Hence  Eb  =  }$  mb-Vn*  +  %  Ibub2 

=  y2  mb  [OH'2  +  kb'2}  ^  ft.-pds. 

Let  the  quantity  in  the  square  bracket  be  denoted  by  Kb2', 
then  evidently  Kb2  may  be  considered  as  the  radius  of  gyration 
of  a  body,  which  if  secured  to  the  link  a  and  having  a  mass  ra& 
would  have  the  same  kinetic  energy  as  the  link  6  has  at  this 
instant.  It  is  evidently  a  very  simple  matter  to  find  Kb  graphi- 
cally since  it  is  the  hypotenuse  of  the  right-angled  triangle  of 
which  one  side  is  OH'  and  the  other  kb\  this  construction  is  shown 
in  Fig.  149. 


Thus  Eb  =  M  nibKbW  ft.-pds, 

Similarly  Ea  =  ^m0Xa2co2  ft.-pds. 

and  Ec  =  ^mcKc2u2  ft.-pds.  ; 

constructions  for  Ka  and  Kc  are  shown  in  the  figure. 
For  the  whole  machine  the  kinetic  energy  is 
E  =  Ea  +  Eb  +  Ec 


+  /'*  +  /'el 
ft.-pds. 


246  THE  THEORY  OF  MACHINES 

A  study  of  these  formulas  and  a  comparison  with  the  work 
just  covered,  shows  that  I'a  is  the  moment  of  inertia  of  the  mass 
with  center  of  gravity  at  0  and  rotating  with  angular  velocity 
co  which  will  have  the  same  kinetic  energy  as  the  link  a  actually 
has;  in  other  words,  Ira  may  be  looked  upon  as  the  reduced  mo- 
ment of  inertia  of  the  link  a,  while  similar  meanings  may  be 
attached  to  /'&  and  I'c.  Note  that  I'a  and  Ia  differ  because  the 
former  is  the  inertia  of  the  corresponding  mass  with  center  of 
gravity  at  0,  whereas  Ia  is  the  moment  of  inertia  about  the 
center  of  gravity  G  of  the  actual  link.  The  quantity  J  is,  on 
the  same  basis,  the  reduced  inertia  of  the  entire  machine,  by 
which  is  meant  that  the  kinetic  energy  of  the  machine  is  the 
same  as  if  it  were  replaced  by  a  single  mass  with  center  of  gravity 
at  0,  and  having  a  moment  of  inertia  J  about  0,  this  mass  rotat- 
ing at  the  angular  speed  of  the  primary  link.  It  will  be  readily 
understood  that  J  differs  for  each  position  of  the  machine  and  is 
also  a  function  of  the  form  and  weight  of  the  links. 

The  foregoing  method  of  reduction  is  of  the  greatest  importance 
in  solving  the  problems  under  consideration,  because  it  makes  it 
possible  to  reduce  any  machine,  no  matter  how  complex,  down 


FIG.  150. 

to  a  single  mass,  rotating  with  known  speed,  about  a  fixed  center, 
so  that  the  kinetic  energy  of  the  machine  is  readily  found  from 
the  drawing. 

204.  Application  to  Reciprocating  Engine. — The  method  may 
be  further  illustrated  in  the  common  case  of  the  reciprocating 
engine,  which  in  addition  to  the  turning  pairs  contains  also  a 
sliding  pair.  The  mechanism  is  shown  in  Fig.  150  and  the 
same  notation  is  employed  as  was  used  in  the  previous  case,  and 
the  only  peculiarity  about  the  mechanism  is  the  treatment  of 
the  link  c. 

The  link  c  has  a  motion  of  translation  only  and  therefore 
coc  =  0  and  /cwc2  =  0  so  that  the  kinetic  energy  of  the  link  is 


SPEED  FLUCTUATIONS  IN  MACHINERY        247 

Ec  =  }4  mc-  vcz  =  M  mc  OQ-'2o>2  or  7C'  =  mcOQ'2  since  the 
point  Q  has  the  same  linear  velocity  as  all  points  in  the  link. 
The  remainder  of  the  machine  is  treated  as  before. 

Lack  of  space  prevents  further  multiplication  of  these  illus- 
trations, but  it  will  be  found  that  the  method  is  easily  applied 
to  any  machine  and  that  the  time  required  to  work  out  the  values 
of  J  for  a  complete  cycle  is  not  very  great. 

SPEED  FLUCTUATIONS 

205.  Conditions  Affecting  Speed  Variations. — One  of  the  most 
useful  applications  of  the  foregoing  theory  is  to  the  determination 
of  the  proper  weight  of  flywheel  to  suit  given  running  conditions 
and  to  prevent  undue  fluctuations  in  speed  of  the  main  shaft  of 
a  prime  mover.  Usually  the  allowable  speed  variations  are  set 
by  the  machine  which  the  engine  or  turbine  or  other  motor  is 
driving  and  these  variations  must  be  kept  within  very  narrow 
limits  in  order  to  make  the  engine  of  value.  When1  a  dynamo  is 
being  driven,  for  example,  fluctuations  in  speed  affect  the  lights, 
causing  them  to  flicker  and  to  become  so  annoying  in  certain  cases 
that  they  are  useless.  The  writer  has  seen  a  particularly  bad 
case  of  this  kind  in  a  gas  engine  driven  generator.  If  alternators 
are  to  be  run  in  parallel  the  speed  fluctuation  must  be  very  small 
to  make  the  arrangement  practicable. 

In  many  rolling  mills  motors  are  being  used  to  drive  the  rolls 
and  in  such  cases  the  rolls  run  light  until  a  bar  of  metal  is  put  in, 
and  then  the  maximum  work  has  to  be  done  in  rolling  the  bar. 
Thus,  in  such  a  case  the  load  rises  suddenly  from  zero  to  a  maxi- 
mum and  then  falls  off  again  suddenly  to  zero.  Without  some 
storage  of  energy  this  would  probably  cause  damage  to  the  motor 
and  hence  it  is  usual  to  attach  a  heavy  flywheel  somewhere 
between  the  motor  and  the  rolls,  this  flywheel  storing  up  energy 
as  it  is  being  accelerated  after  a  bar  has  passed  through  the  rolls, 
and  again  giving  out  part  of  its  stored-up  energy  as  the  bar  enters 
and  passes  through  the  rolls.  The  electrical  conditions  determine 
the  allowable  variation  in  speed,  but  when  this  is  known,  and  also 
the  work  required  to  roll  the  bar  and  the  torque  which  the  motor 
is  capable  of  exerting  under  given  conditions,  then  it  is  necessary 
to  be  able  to  determine  the  proper  weight  of  flywheel  to  keep  the 
speed  variation  within  the  set  limits. 

In  the  case  of  a  punch  already  mentioned,  the  machine  runs 


248  THE  THEORY  OF  MACHINES 

light  for  some  time  until  a  plate  is  pushed  in  suddenly  and  the 
full  load  is  thrown  on  the  punch.  If  power  is  being  supplied  by 
a  belt  a  flywheel  is  also  placed  on  the  machine,  usually  on  the 
shaft  holding  the  belt  pulley,  this  flywheel  storing  up  energy  while 
the  machine  is  light  and  assisting  the  belt  to  drive  the  punch 
through  the  plate  when  a  hole  is  being  punched.  The  allowable 
percentage  of  slip  of  the  belt  is  usually  known  and  the  wheel  must 
be  heavy  enough  to  prevent  this  amount  of  slip  being  exceeded. 

The  present  discussion  is  devoted  to  the  determination  of 
the  speed  fluctuations  with  a  given  machine,  and  the  investiga- 
tion will  enable  the  designer  to  devise  a  machine  that  will  keep 
these  fluctuations  within  any  desired  limits,  although  the  next 
chapter  deals  more  particularly  with  this  phase. 

206.  Determination  of  Speed  Variation  in  Given  Machine. — 
Let  EI  and  E%  be  the  kinetic  energies,  determined  as  already 
explained,  of  any  machine  at  the  beginning  and  end  of  a  certain 
interval  of  time  corresponding  with  a  definite  change  of  posi- 
tion of  the  parts.  Then  the  gain  in  energy,  E%  —  EI,  during  the 
interval  under  consideration  represents  the  difference  between 
the  energy  supplied  with  the  working  fluid  and  the  sum  of  the  fric- 
tion of  the  machine  and  of  the  work  done  at  the  main  shaft  on 
some  other  machine  or  object  during  the  same  interval,  because 
the  kinetic  energy  of  the  machine  can  only  change  from  instant 
to  instant  if  the  work  done  by  the  machine  differs  from  the  work 
done  on  it  by  the  working  fluid.  In  order  to  simplify  the  problem 
friction  will  be  neglected,  or  assumed  included  in  the  output. 

Consideration  will  show  that  E2  —  EI  will  be  alternately  posi- 
tive and  negative,  that  is,  during  the  cycle  of  the  machine  its 
kinetic  energy  will  increase  to  a  maximum  and  then  fall  again 
to  a  minimum  and  so  on.  As  long  as  the  kinetic  energy  is  in- 
creasing the  speed  of  the  machine  must  also  increase  in  general, 
so  that  the  speed  will  be  a  maximum  just  where  the  kinetic  energy 
begins  to  decrease,  and  conversely  the  speed  will  be  a  minimum 
just  where  the  kinetic  energy  begins  to  increase  again.  But  the 
kinetic  energy  of  the  machine  will  increase  just  so  long  as  the 
energy  put  into  the  machine  is  greater  than  the  work  done  by  it 
in  the  same  time;  hence  the  maximum  speed  occurs  at  the  end  of 
any  period  in  which  the  input  to  the  machine  exceeds  its  output 
and  vice  versa. 

The  method  of  computing  this  speed  fluctuation  will  now  be 
considered. 


SPEED  FLUCTUATIONS  IN  MACHINERY        249 

It  has  already  been  shown  that  the  kinetic  energy  of  the  machine 
is  given  by 

E  =  KJco2  ft.-pds. 

from  which  there  is  obtained  by  differentiation1 

5E  =  %  {2J.  co.5co  +  co2.5J|  ft.-pds. 
or 

5E  -co2.5J 


r 

Jco 

where  6co  is  the  change  in  speed  in  radians  per  second  in  the 
interval  of  time  in  which  the  gain  of  energy  of  the  machine  is 
5E  and  that  in  J  is  dJ.  Of  course,  any  of  these  changes  may  be 
positive  or  negative  and  they  are  not  usually  all  of  the  same  sign. 
The  values  of  J  and  co  used  in  the  formula  may,  without  sensible 
error,  be  taken  as  those  at  the  beginning  or  end  of  the  inter- 
val or  as  the  average  throughout  the  interval,  the  latter  being 
preferable. 

207.  Approximate  Value  of  Speed  Variation.  —  The  calculation 
is  frequently  simplified  by  making  an  approximation  on  the 
assumption  that  the  variation  in  J  may  be  neglected,  i.e.,  that 
8J  =  fc0  The  writer  has  not  found  that  there  is  enough  saving 
in  time  in  the  work  involved  to  make  this  approximation  worth 
while,  but  since  it  is  often  assumed,  it  is  placed  here  for  con- 
sideration and  a  slightly  different  method  of  deducing  the 
resulting  formula  is  given.  Let  Ei,  Ez,  coi  and  co2  have  the 
meanings  already  assigned,  at  the  beginning  and  end  of  the 
interval  of  time  and  let  the  reduced  moment  /  be  considered 
constant. 

1  To  those  not  familiar  with  the  calculus  the  following  method  may  be  of 
value. 

Let  E,  J  and  co  be  the  values  of  the  quantities  at  the  beginning  of  the 
interval  of  time  and  E  +  dE,  J  +  8  J  and  co  -}-  5co,  the  corresponding  values 
at  the  end  of  the  same  interval. 
Then 

E  =  l^Jco2 
E  +  5E  =%(J  +  6J")  (co  +  5co)2 

=  J^C/co8  +  2J.co5co  +  co25j) 

where  in  the  multiplication  such  terms  as  (5co)2,  and  Sj.Sco  are  neglected  as 
being  of  the  second  order  of  small  quantities. 
By  subtraction,  then, 

E  +  8E-E=*dE  =  %  {2J"co5w  +  co25J"j  as  above. 


250 
Then 

or 


THE  THEORY  OF  MACHINES 

?    -    CO!2) 


where  co  has  been  written  for        0      1,  a  substitution  which  causes 

a 

little  inaccuracy  in  practice. 
Therefore  co2  —  coi  = 


—  E\ 


«7co 


or 


000  —  -7-. 

t/CO 


This  is  the  same  result  as  would  have  been  obtained  from  the 
former  formula  by  making  5J  =  0. 

208.  Practical  Application  to  the  Engine. — The  meanings  of 
the  different  quantities  can  best  be  explained  by  an  example 


FIG.  151. 

which  will  now  be  worked  out.  The  steam  engine  has  been 
selected,  because  all  the  principles  are  involved  and  the  method 
of  selecting  the  data  in  this  case  may  be  rather  more  readily 
understood.  The  computations  have  all  been  made  by  the  exact 
formula,  which  takes  account  of  variations  in  J. 

Consider  the  double-acting  engine,  which  is  shown  with  the 
indicator  diagrams  in  Fig.  151;  it  is  required  to  find  the  change  of 
speed  of  the  crank  while  passing  from  A  to  B.  Friction  will  be 
neglected. 

For  simplicity,  it  will  be  assumed  that  the  engine  is  driving  a 


SPEED  FLUCTUATIONS  IN  MACHINERY        251 

turbine  pump  which  offers  a  uniform  resisting  turning  moment 
and  hence  the  work  done  by  the  engine  during  any  interval  is 
proportional  to  the  crank  angle  passed  through  in  the  given 
interval.  If  the  work  done  per  revolution  as  computed  from 
the  diagram  is  W  ft.-pds.,  then  the  work  done  by  the  engine  during 

a     __  n 

the  interval  from  A  to  B  will  be     QAn    W  ft.-pds.     To  make  the 

OOU 

case  as  definite  as  possible  suppose  that  02  —  0i  =  18°;  then  the 
work  done  by  the  engine  will  be  Ho^  ft.-pds. 

209.  Output  and  Input  Work.  —  Again,  let  AI  and  A2  represent 
the  areas  in  square  inches  of  the  head  end  and  the  crank  end  of 
the  cylinder  respectively,  li  and  Z2  being  the  lengths  of  the  cor- 
responding indicator  diagrams  in  inches.  The  stroke  of  the  pis- 
ton is  taken  as  L  feet  and  the  indicator  diagrams  are  assumed 
drawn  to  scale  s  pds.  per  square  inch  =  1  in.  in  height.  With 
these  symbols  the  work  represented  by  each  square  inch  on  the 

diagram  is  sAi  >-  ft.-pds.  for  the  head-end  and  8^27-  ft.-pds.  for  the 
l>i  L% 

crank-end  diagram. 

Now  suppose  that  during  the  crank's  motion  from  A  to  B  the 
area  of  the  head-end  diagram  reckoned  above  the  zero  line  is 
a\  sq.  in.,  see  Fig.  151,  and  the  corresponding  area  for  the  crank- 
end  diagram  a2  sq.  in.  Then  the  energy  delivered  to  the  engine 
by  the  steam  during  the  interval  is 


ij  --     zzf-  ft.-pds. 
LI  L2 

while  the  work  done  by  the  engine  is 

W 
20 

Note  that  the  work  W  is  the  total  area  of  the  two  diagrams  in 
square  inches  multiplied  by  their  corresponding  constants  to 
bring  the  quantities  to  foot-pounds. 

Then  the  input  work  exceeds  the  output  by 

L  L  W 


*. 

i  y  --    ^z-j  --  ™  ft.-pds. 
LI  LZ       ^U 

which  amount  of  energy  must  be  stored  up  in  the  moving  parts 
during  the  interval.  That  is,  the  gain  in  energy  during  the 
period  is 

E2  —  EI  =  aisAi  7  --  a2sAz  -,  --  o~  ft.-pds. 


so  that  the  gain  in  energy  is  thus  known. 


252  THE  THEORY  OF  MACHINES 

Again,  the  method  described  earlier  in  the  chapter  enables  the 
values  of  Ji  and  J2  to  be  found  and  hence  the  value  of  J2  —  J\. 
Substituting  these  in  the  formula, 

dE  -  Kco25J 
5co  =  -      — 7 — 

J  CO 

the  gain  in  angular  velocity  is  readily  found.  The  values  are 
#2  -  Ei  =  dE,  J2  -  Jl  =  8J,  MGA  +  Ji)  =  J  and  for  co  no 
error  will  result  in  practice  by  using  the  mean  speed  of  rotation 
of  the  crank. 

During  a  complete  revolution  the  values  of  5co  will  sometimes 
be  positive  and  sometimes  negative,  and  in  order  that  the  engine 
may  maintain  a  constant  mean  speed  the  algebraic  sum  of  these 
must  be  zero.  Should  the  algebraic  sum  for  a  revolution  be 
positive,  the  conclusion  would  be  that  there  is  a  gain  in  the  mean 
speed  during  the  revolution,  that  is  the  engine  would  be  steadily 
gaining  in  speed,  whereas  it  has  been  assumed  that  the  governor 
prevents  this. 

210.  Numerical  Example  on  Single -cylinder  Engine. — A  nu- 
merical example  taken  from  an  actual  engine  will  now  be  given. 


FIG.  152. 

The  engine  used  in  this  computation  had  a  cylinder  12>f  6  in. 
diameter  with  a  piston  rod  1%  in.  diameter  and  a  stroke  of  30  in. 
The  connecting  rod  was  90  in.  long,  center  to  center,  weighed 
175  Ib.  and  had  a  radius  of  gyration  about  its  center  of  gravity 
of  31.2  in.  The  piston,  crosshead  and  other  reciprocating  parts 
weighed  250  Ib.,  while  the  flywheel  weighed  5,820  Ib.  and  had  a 
moment  of  inertia  about  the  shaft  of  2,400,  using  pound  and 
foot  units.  The  mean  speed  of  rotation  was  86  revolutions  per 
minute. 

Using  the  notation  employed  in  the  earlier  discussion,  the  data 
may  be  set  down  as  follows: 

a  =  1.25  ft.,   6  =  7.5  ft.,   fa  =  2.60  ft.,  Ia  =  2,400  pd.(ft.)2, 
ma  =  181,  mb  =  5.44,  mc  =  7.78. 

™*  27m      2  X  T  X  8£ 

Ine  speed  oy=  ~^-    --  —        =  9  radians  per  second. 


SPEED  FLUCTUATIONS  IN  MACHINERY        253 


Using  the  above  data  the  following  quantities  were  measured 
directly  from  the  drawing,  Fig.  152: 


e 

degrees 

b' 

feet 

OH' 

feet 

k'b 

feet 

Kb 
feet 

OQ' 

feet 

36 

1.017 

0.935 

0.352 

1.00 

0.84 

54 

0.741 

1.123 

0.257 

1.15 

1.11 

From  these  the  following  quantities  are  obtained  by  computa- 
tion: 


B 

degrees 

I'b  =*=  mb.Kb- 

,,-„„.<*, 

2 

J  = 

la+I'b+P, 

SJ 

36 

5.442 

5.488 

2,400 

2,410.9 

+5,9 

54 

7.200 

9.578 

2,400' 

2,416.8 

120 


GO 


Zero      Line 

FIG.  153. 


a  g  =.035  Sq.  In. 


Thus,  during  the  18°  under  consideration  there  is  a  gain  in 
the  reduced  inertia  of  5.9,  although  as  the  complete  table  given 
later  on  shows,  there  is  a  loss  in  other  parts  of  the  revolution. 

The  indicator  diagrams  for  the  engine  are  shown  in  Fig.  153 
and  the  areas  corresponding  to  the  crank  motion  considered  are 


254 


THE  THEORY  OF  MACHINES 


shown  hatched  and  marked  «i  and  a2.  These  areas  were  meas- 
ured on  the  original  diagrams  which  were  drawn  to  60-  pd.  scale, 
although  these  have  been  somewhat  reduced  in  reproduction. 

Data  for  computations  from   the  indicator  diagrams  are   as 
follows: 

Cylinder  areas:  Head  end,  AI  =  114.28  sq.  in.     Crank  end, 
Az  =  111.52  sq.  in. 

Diagram  lengths:  Head-end  diagram,  li  =  3.55  in.     Cranknend 
diagram,  Z2  =  3.58  in. 

Stroke  of  piston,  L  =  2.5  ft. 
Hence,  each  square  inch  on  the  diagrams  represents 

L  25 

sAir  =  60  X  114.28  X  5-^  =  4,829  ft.-pds.  for  the  head  end, 

LI  o.OO 

and 

= 


=  60  X  111.52  X 


=  4,673  ft.-pds.  for  the  crank  end. 


The  original  full-sized  diagrams  give  ai  =  0.550  sq.  in.  and 
az  =  0.035  sq.  in.,  from  which  the  corresponding  work  done  will 
be: 

0.550  X  4;829  =  2,656  ft.-pds.  for  the  head  end, 
and 

0.035  X  4,673  =  163  ft.-pds.  for  the  crank  end. 
It  is  assumed  that  the  engine  is  driving  a  turbine  pump  or 
electric  generator  which  offers  a  constant  resisting  torque,  so 

-I  O  -j 

that  the  corresponding  work  output  is  ^7:  =  ^  of  the  total  work 


represented  by  the  two  diagrams,  and  is  1,079  ft.-pds. 
The  quantities  are  set  down  in  the  table  below. 


0 
degrees 

Diagram  areas 

Work  done  on  piston 

Work  done 
by  crank, 
ft.-pds. 

Net  work  pro- 
ducing change  of 
kinetic  energy, 
ft.-pds. 

Head  01, 
sq.  in. 

Crank  02, 
sq.  in. 

Head, 
ft.-pds. 

Crank, 
ft.-pds. 

Total, 
ft.-pds. 

36 
54 

0.550 

0.035 

2,656 

163 

2,493 

1,079 

1,414 

The  total  combined  areas  of  the  two  diagrams  represent  21,584 
ft.-pds.,  and  since  the  speed  was  86  revolutions  per  minute  the 

indicated  horsepower  was  QQ'QQQ  X  86  =  56.2  hp. 

The  quantity  in  the  last  column  is  the  difference  between 
2,493  and  1,079  and  would  evidently  cause  the  machine  to  speed  up. 


SPEED  FLUCTUATIONS  IN  MACHINERY        255 

Then  the  work  available  for  increasing  the  energy  is  1,414 
ft.-pds.  and  this  must  represent  the  gain  in  kinetic  energy  of  the 
machine,  or 

5E  =  +  1,414  ft.-pds. 

The  gain  in  angular  velocity  may  now  be  computed.  The 
average  value  of  J  is 

/  =  K  [2,410.9  +  2,416.8]  =  2,413.8 
hence  J.co  =  2,413.8  X  9  =  21,724.6 

and         K  "2.6J  =    MX  92  X  5.9  =  238.9 

dE  -  H  "25/ 

do)    = 

therefore 

_   1,414  -  238.9 

21,724.6 

=  0.0541  radians  per  second 

which  is  the  gain  in  velocity  during  the  period  considered.  Sim- 
ilarly the  results  may  be  obtained  for  other  periods,  and  thus  for 
the  whole  revolution.  These  results  are  set  down  in  the  table 
given  on  page  257. 

211.  Speed-variation  Diagram. — The  values  of  Sto  thus  ob- 
tained are  then  plotted  on  a  straight-line  base,  Fig.  154,  which 
has  been  divided  into  20  equal  parts  to  represent  each  18°  of 
crank  angle.  If  it  is  assumed  that  the  speed  variation  is  small, 
as  it  always  must  be  in  engines,  then  no  serious  error  will  be  made 
by  assuming  that  these  crank  angles  are  passed  through  in  equal 
times,  and  hence  that  the  base  of  the  diagram  on  which  the  values 
of  5 co  are  plotted  is  also  a  time  base,  equal  distances  along  which 
represent  equal  intervals  of  time. 

If  desired,  the  equal  angle  base  may  be  corrected  for  the  varia- 
tions in  the  velocity,  using  the  values  of  5co  already  found,  so  as 
to  make  the  base  exactly  represent  time  intervals,  but  the  author 
does  not  think  it  worth  the  labor  and  has  made  no  correction 
of  this  kind  on  the  diagram  shown. 

Attention  should  here  be  drawn  to  the  fact  that  the  height  of 
the  original  base  used  for  plotting  the  speed-variation  curve  has 
to  be  chosen  at  random,  but  after  the  curve  has  been  plotted,  it  is 
necessary  to  find  a  line  on  this  diagram  representing  the  mean 
speed  of  rotation,  co  =  9.  This  may  be  readily  done  by  finding 
the  area  under  the  curve  by  a  planimeter,  or  otherwise,  and  then 
locating  the  line  co  =  9  so  that  the  positive  and  negative  areas 


256 


THE  THEORY  OF  MACHINES 


SPEED  FLUCTUATIONS  IN  MACHINERY        257 


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17 


258  THE  THEORY  OF  MACHINES 

between  this  new  line  and  the  velocity-variation  curve  are  equal. 
It  is  to  be  remembered  that  the  computation  gives  the  gain  in 
velocity  in  each  interval,  and  the  result  is  plotted  from  the  end 
of  the  curve,  and  not  from  the  base  line  in  each  case. 

212.  Angular-space  Variation. — Now  since  the  space  traversed 
is  the  product  of  the  corresponding  velocity  and  time,  the  angular- 
space  variation,  56  in  radians,  is  found  by  multiplying  the  value 
of  d  co  by  the  time  t  in  seconds  required  to  turn  the  crank  through 
the  corresponding  18°,  that  is 

56  =  t.5u  radians. 

But  t.5u  is  evidently  an  area  on  the  curve  of  angular-velocity 
variation,  so  that  the  angular  space  variation  in  radians  up  to 
any  given  crank  angle,  say  54°,  is  simply  the  area  under  the 
angular-velocity  variation  curve  up  to  this  point,  the  area  being 
taken  with  the  mean  angular  velocity  as  a  base  and  not  with  the 
original  base  line.  In  this  case  the  area  between  the  mean  speed 
line,  co  =  9,  and  the  speed-variation  curve,  from  0  to  54°,  when 
reduced  to  proper  units,  represents  0.275  radian  as  plotted  in 
the  lower  curve  of  Fig.  154. 

The  upper  curve  shows  that  the  minimum  angular  velocity 
was  8.922  radians  per  second;  while  the  maximum  was  9.063 
radians  per  second,  a  variation  of  0.141  radian  per  second,  or 
1.57  per  cent. 

The  lower  curve  shows  the  angular  swing  of  the  flywheel  about 
its  mean  position,  and  shows  that  the  total  swing  between  the 
two  extremes  was  0.58°,  although  the  swing  from  the  mean  posi- 
tion would  be  only  about  one-half  of  this. 

The  complete  computations  which  have  been  given  here  in 
full  for  an  engine,  will,  it  is  hoped,  clearly  illustrate  the  method 
of  procedure  to  be  followed  in  any  case.  The  method  is  not  as 
lengthy  as  would  appear  at  first,  and  the  results  for  an  engine 
may  be  quickly  obtained  by  the  use  of  a  slide  rule  and  drafting 
board. 

In  the  case  of  engines,  all  moving  parts  have  relatively  high 
velocity,  and  it  is  generally  advisable  to  take  account  of  the 
variations  in  the  reduced  inertia,  J.  In  other  machines,  such  for 
example  as  a  belt-driven  punch,  all  parts  are  very  slow-moving 
with  the  exception  of  the  shaft  carrying  the  belt  pulleys  and  fly- 
wheel, and  in  such  a  case  it  is  only  necessary  to  take  account  of 
the  inertia  of  the  high-speed  parts.  Wherever  the  parts  are  of 


SPEED  FLUCTUATIONS  IN  MACHINERY        259 


large  size  or  weight  or  run  at  high  speed,  account  must  be  taken 
of  their  effect  on  the  machine. 


jg   esBaioui  o 

puooas 


{3       8SBQJ09Q 


Frequently  only  the  angular-velocity  variation  is  required,  but 
usually  the  space  variation  is  also  necessary,  as  in  the  case  of 
alternators  which  are  to  work  in  parallel. 


260  THE  THEORY  OF  MACHINES 

213.  Factors  Affecting  the  Speed  Fluctuations. — A  general  dis- 
cussion has  been  given  earlier  in  this  chapter  of  the  factors  that 
affect  the  magnitude  of  the  speed  fluctuations  in  machinery  and 
as  an  illustration  here  Fig.  155  has  been  drawn.  This  figure 
shows  three  speed-fluctuation  curves  for  the  engine  just  referred 
to,  and  for  the  same  indicator  diagrams  as  are  shown  in  Fig.  153, 
but  in  each  case  the  engine  is  used  for  a  different  purpose.  The 
curve  in  the  plain  line  is  an  exact  copy  of  the  upper  curve  in 
Fig.  154  and  represents  the  fluctuations  which  occur  when  the 
engine  is  direct-coupled  to  an  electric  generator,  the  total  fluc- 
tuation being  0.14  radian  per  second  or  about  1.57  per  cent. 

The  dotted  curve  corresponds  to  a  water  pump  connected  in 
tandem  with  the  engine,  a  common  enough  arrangement,  al- 
though the  piston  speed  is  rather  too  high  for  this  class  of  work. 
The  speed  fluctuation  here  would  be  less  than  before,  amounting 
to  0.123  radian  or  about  1.37  per  cent.,  this  being  due  to  the 
fact  that  the  unbalanced  work  is  not  so  great  in  this  class  of  re- 
sistance as  in  the  generator. 

The  broken  line  corresponds  to  an  air-compressor  cylinder  in 
tandem  with  the  steam  cylinder  and  the  resulting  variation  is 
0.305  radian  per  second  or  3.38  per  cent.,  which  is  over  twice 
as  much  as  the  first  case. 

QUESTIONS  ON  CHAPTER  XIII 

1.  A  12-in.  round  cast-iron  disk  2  in.  thick  has  a  linear  velocity  of  88  ft. 
per  second;  find  its  kinetic  energy.     What  would  be  its  kinetic  energy  if  it 
also  revolved  at  100  revolutions  per  minute? 

2.  A  straight  steel  rod  2  ft.  long,  1^  in.  diameter,  rotates  about  an  axis 
normal  to  its  center  line,  and  6  in.  from  its  end,  at  50  revolutions  per  minute. 
What  is  its  kinetic  energy? 

3.  Find  the  kinetic  energy  of  a  wheel  12  in.  diameter,  density  2. 14,  at  500 
revolutions  per  minute. 

4.  What  is  the  kinetic  energy  of  a  cast-iron  wheel  3  ft.  diameter,  1%  in. 
thick,  rolling  on  the  ground  at  8  miles  per  hour? 

6.  If  the  side  rod  of  a  locomotive  is  5  ft.  long  and  of  uniform  section 
2>£  by  5  in.,  with  drivers  60  in.  diameter,  and  a  stroke  of  24  in.,  find  the 
kinetic  energy  of  the  rod  in  the  upper  and  lower  positions. 

6.  Show  how  to  find  the  kinetic  energy  of  the  tool  sliding  block  of  the 
Whitworth  quick-return  motion. 

7.  Suppose  the  wheel  in  question  3  is  a  grinder  used  to  sharpen  a  tool  and 
that  its  speed  is  decreased  in  the  process  to  450  revolutions  in  1  sec.;  what  is 
the  change  in  kinetic  energy? 

8.  Plot  the  speed  and  angular  velocity-variation  curves  for  two  engines  like 
that  discussed  in  the  text  with  cranks  at  90°,  only  one  flywheel  being  used. 

9.  Repeat  the  above  with  cranks  at  180°. 

\J 


CHAPTER  XIV 
THE  PROPER  WEIGHT  OF  FLYWHEELS 

214.  Purpose  of  Flywheels. — In  the  preceding  chapter  a  com- 
plete discussion  has  been  given  as  to  the  causes  of  speed  fluc- 
tuations in  machinery  and  the  method  of  determining  the  amount 
of  such  fluctuation.  In  many  cases  a  certain  machine  is  on  hand 
and  it  is  the  province  of  the  designer  to  find  out  whether  it  will 
satisfy  certain  conditions  which  are  laid  down.  This  being  the 
case  the  problem  is  to  be  solved  in  the  manner  already  discussed, 
that  is,  the  speed  fluctuation  corresponding  to  the  machine  and 
its  methods  of  loading  are  to  be  determined. 

Frequently,  however,  the  converse  problem  is  given,  that  is, 
it  is  required  to  design  a  machine  which  will  conform  to  certain 
definite  conditions;  thus  a  steam  engine  may  be  required  for 
driving  a  certain  machine  at  a  given  mean  speed  but  it  is  also 
stipulated  that  the  variation  in  speed  during  a  revolution  must 
not  exceed  a  certain  amount.  Or  a  motor  may  be  required  for 
driving  the  rolls  in  a  rolling  mill,  the  load  in  such  a  case  varying 
so  enormously,  that,  if  not  compensated  for  would  cause  great 
fluctuations  in  speed  in  the  motor,  which  fluctuations  might  be  so 
bad  as  to  prevent  the  use  of  the  motor  for  the  purpose.  In  a 
punch  or  shear  undue  fluctuation  in  speed  causes  rapid  destruc- 
tion of  the  belt.  In  all  the  above  and  similar  cases  these  varia- 
tions must  be  kept  within  certain  limits  depending  upon  the 
machine. 

In  all  machines  certain  dimensions  are  fixed  by  the  work  to 
be  done  and  the  conditions  of  loading,  and  are  very  little  affected 
by  the  speed  variations.  Thus,  the  diameter  of  the  piston  of  an 
engine  depends  upon  the  power,  pressure,  mean  speed,  etc.,  and 
having  determined  the  diameter,  the  thickness  and  therefore  the 
weight  is  fixed  by  the  consideration  of  strength  almost  exclu- 
sively; the  same  thing  is  largely  true  regarding  the  crosshead,  con- 
necting rod  and  other  parts,  the  dimensions,  weights  and  shapes 
being  independent  of  the  speed  fluctuations.  Similar  statements 
may  be  made  about  the  motor,  its  bearings,  armature,  etc.,  being 
fixed  by  the  loading,  and  in  a  punch  the  size  of  gear  teeth 
and  other  parts  are  also  independent  of  the  speed  fluctuation. 

261 


262  THE  THEORY  OF  MACHINES 

Each  of  these  machines  contains  also  a  flywheel,  the  dimen- 
sions of  which  depend  on  the  speed  variations  alone  and  not  upon 
the  power  or  pressures  as  do  the  other  parts.  The  function  of 
the  flywheel  is  to  limit  these  variations;  thus  on  a  given  size 
and  make  of  engine  the  weight  of  flywheel  will  vary  greatly  with 
the  conditions  of  working;  in  some  cases  the  wheel  would  be  very 
heavy,  while  in  other  cases  there  might  be  none  at  all  on  the  same 
engine. 

Ordinarily  the  flywheel  is  made  heavy  and  run  with  as  high  a 
rim  speed  as  is  deemed  safe;  in  slow-revolving  engines  the 
diameter  is  generally  large,  while  in  higher-speed  engines  the 
diameter  is  smaller,  as  in  automobile  engines,  etc.  The  present 
chapter  is  devoted  to  the  method  of  determining  the  dimensions 
of  flywheel  necessary  to  keep  the  speed  fluctuations  in  a  given 
case  within  definitely  fixed  limits. 

Referring  to  Chapter  XIII,  Sec.  203,  the  kinetic  energy  of  a 
machine  is  given  by  the  equation  E  =  %Ju2,  where  J  is  the  re- 
duced inertia  found  as  described  therein.  The  method  of 
obtaining  E  has  also  been  fully  explained;  it  depends  upon  the 
input  and  output  of  the  machine,  such,  for  example,  as  the  indi- 
cator and  load  curves  for  a  steam  engine.  E  and  J  are  thus 
assumed  known  and  the  above  equation  may  then  be  solved  for 

W 
co,  thus,  j/2  o>2  =  -  j.- 

215.  General  Discussion  of  the  Method  Used. — In  order  that 
the  matter  may  be  most  clearly  presented  it  will  be  simplest  to 
apply  it  to  one  particular  machine  and  the  one  selected  is  the 
reciprocating  engine,  because  it  contains  both  turning  and  sliding 
elements  and  gives  a  fairly  general  treatment.  In  almost  all 
machines  there  are  certain  parts  which  turn  at  uniform  speed 
about  a  fixed  center  and  which  have  a  constant  moment  of  in- 
ertia, such  as  the  crank  and  flywheel  in  an  engine,  while  other 
parts,  such  as  the  connecting  rod,  piston,  etc.,  have  a  variable 
motion  about  moving  centers  and  a  correspondingly  variable 
reduced  moment  of  inertia;  the  table  in  the  preceding  chapter 
illustrates  this.  It  will  be  convenient  to  use  the  symbol  Ja  to 
represent  the  moment  of  inertia  of  the  former  parts,  while  J& 
represents  that  of  the  latter,  and  thus  Ja  is  constant  for  all 
positions  of  the  machine,  and  Jb  is  variable.  The  total  reduced 
inertia  of  the  machine  is  J  =  Ja  +  Jb-  Both  of  these  quantities 


THE  PROPER  WEIGHT  OF  FLYWHEELS 


263 


J0  and  Jb  are  independent  of  the  speed  of  rotation  and  depend 
only  upon  the  mass  and  shape  of  the  links,  that  is  upon  the  rela- 
tive distribution  of  the  masses  about  their  centers  of  gravity. 

Suppose  now  that  for  any  machine  the  values  of  J  are  plotted 
on  a  diagram  along  the  ff-axis^tho  nrrMrmf.pa  nf  which  diagram 
represent  the  corresponding  value  of  the  energy  \E±  this  will  give 
a  diagram  of  the  general  shape  shown  at  Fig.  156.  where  the  curve 


E' 


H 


\      8 


ii 


R 


FIG.  156. 


represents  J  for  the  corresponding  value  of  E  shown  on  the 
vertical  line.1 

Looking  now  at  the  figure  KFGHK,  it  is  evident  from  construc- 
tion that  its  width  depends  on  the  values  of  J  at  the  instant  and 
is  thus  independent  of  the  speed.  Also,  the  height  of  this  figure 
depends  on  the  difference  between  the  work  put  into  the  machine 
and  the  work  delivered  by  the  machine  during  given  intervals, 
that  is,  it  will  depend  on  such  matters  as  the  shapes  of  the  indi- 
cator and  load  curves.  The  shape  of  the  input  work  diagrams 
within  certain  limits  depends  on  whether  the  machine  is  run  by 
gas  or  steam,  and  on  whether  it  is  simple  or  compound,  etc.,  but 
for  a  given  engine  this  is  also,  generally  speaking,  independent  of 
the  speed :  the  load  curve  will,  of  course,  depend  on  what  is  being 
driven,  whether  it  is  dynamo,  compressor,  etc.  Thus  the  height 
of  the  figure  is  also  independent  of  the  speed. 

1  This  form  of  diagram  appears  to  be  due  to  WITTENBAUER;  see  "Zeit- 
schrift  des  Vereines  deutscher  Ingenieure"  for  1905. 


264  THE  THEORY  OF  MACHINES 

It  will  further  be  noted  that  the  shape  of  the  figure  does  not 
depend  on  Ja,  which  is  constant  for  a  given  machine,  but  only 
on  the  values  of  the  variable  Jbj  hence  the  shape  of  this  figure 
will  be  independent  of  the  weight  of  the  flywheel  and  speed,  in  so 
far  as  the  input  and  load  curves  are  independent  of  the  speed, 
depending  solely  on  the  reciprocating  masses,  the  connecting 
rod,  the  input-work  diagrams  and  the  load  curves. 

Now  draw  from  0  the  two  tangents,  OF  and  OH,  to  KFGH, 
touching  it  at  F  and  H  respectively,  then  for  OH  the  energy 

Ei  =  HH',  and  Jl  =  OH'  and  (Sec.  203),  JW  =  ^  =  tan  «i, 

J  i 

and  since  «i  is  the  least  value  such  an  angle  can  have  it  is  evident 
that  toi  is  the  minimum  speed  of  the  engine.  Similarly,  E%  =  FFf 

F 
and  <72  =  OF',  and  J^a>22  =  -=-  =  tan  .0:2  and  hence,  co2  would  be 

t/2 

the  maximum  speed  of  the  engine,  since  oti  is  the  maximum  value 
of  a. 

216.  Dimensions  of  the  Flywheel. — Suppose  now  that  it  is 
required  to  find  the  dimensions  of  a  flywheel  necessary  for  a  given 
engine  which  is  to  be  used  on  a  certain  class  of  service,  the  mean 
speed  of  rotation  being  known.  The  class  of  service  will  fix  the 
variations  allowable  and  the  mean  speed ;  in  engines  driving  alter- 
nators for  parallel  operation  the  variation  must  be  small,  while 
in  the  driving  of  air  compressors  and  plunger  pumps  very  much 
larger  variations  are  allowable.  Thus,  the  class  of  service  fixes 
the  speed  variation  o>2  ~  coi  radians  per  second,  and  the  mean 

speed  co  =  -  ~ —  is  fixed  by  the  requirements  of  the  output. 
Zi 

Experience  enables  the  indicator  diagrams  to  be  assumed  with 
considerable  accuracy  and  the  load  curve  will  again  depend  on 
what  class  of  work  is  being  done. 

The  only  part  of  the  machine  to  be  designed  here  is  the  fly- 
wheel, and  as  the  other  parts  are  known,  and  the  indicator  and 
load  curves  are  assumed,  the  values  of  E  and  /&  are  found  as 
explained  in  Chapter  XIII  and  the  E  —  Jb  curve  is  drawn  in. 
In  plotting  this  curve  the  actual  value  of  E  is  not  of  importance, 
but  any  point  may  arbitrarily  be  selected  as  a  starting  point  and 
then  the  values  of  5E,  or  the  change  in  E,  and  Jb  will  alone  give 
the  desired  curve.  Thus,  in  Fig.  156  the  diagram  KFGH  has 
been  so  drawn  and  it  is  to  be  observed  that  the  exact  position 
of  this  figure  with  regard  to  the  origin  0  is  unknown  until  Ja  is 


THE  PROPER  WEIGHT  OF  FLYWHEELS         265 

known,  but  it  is  Ja  that  is  sought.  A  little  consideration  will 
show,  however,  that  an  axis  E'Oi  may  be  selected  and  used  as 
the  axis  for  plotting  Jb,  values  of  which  may  be  laid  off  to  the 
right. 

Further,  any  horizontal  axis  0'  —  Jb  may  be  selected,  and  for 
any  value  of  Jb  a  point  may  be  arbitrarily  selected  to  represent 
the  corresponding  value  of  E  and  the  meaning  of  this  point  may 
be  later  determined.  Having  selected  the  first  point,  the  remain- 
ing points  are  definitely  fixed,  since  the  change  in  E  corresponding 
to  each  change  in  Jb  is  known.  Thus,  the  curve  may  be  found 
in  any  case  without  knowing  Ja  or  the  speed,  but  the  origin  0  has 
its  position  entirely  dependent  upon  both,  and  cannot  be  deter- 
mined without  knowing  them.  Thus  the  correct  position  of  the 
axes  of  E  and  J  are  as  yet  unknown,  although  their  directions 
are  fixed. 

Having  settled  on  coi  and  co2,  two  lines  may  be  drawn  tangent 
to  the  figure  at  H  and  F  and  making  the  angles  a\  and  «2  respec- 
tively, with  the  direction  0'  —  J&,  where  tan  a\  =  J^coi2  and 
tan  a2  =  MW22.  The  intersection  of  these  two  lines  gives  0  and 
hence  the  axis  of  E,  so  that  the  required  moment  of  inertia  of 
the  wheel  may  be  scaled  from  the  figure,  since  Ja  =  00\.  It 
should,  however,  be  pointed  out  that  if  the  position  of  the  axis 
of  E  is  known,  and  also  the  mean  speed  co,  it  is  not  possible  to 
choose  coi  and  co2  at  will,  for  the  selection  of  either  E  or  the  speeds 
will  determine  the  position  of  0.  In  making  a  design  it  is  usual 

to  select  oj  and  —  -  —  —  ,  which  give   coi  and  co2,  and  from  the 

CO 

chosen  values  to  determine  the  position  of  0  and  hence  the  axes  of 
E  and  J.     The  mean  speed  o>  corresponds  with  the  angle  a. 
Draw  a  line  NMLR  perpendicular  to  OJ,  close  to  the  E  —  J 

T  /? 

diagram  but  in  any  convenient  position.     Then          ~  tan  ai, 


NR  A 

jyn  =  tan  a2  and  -^  =  tan  a,  so  that  on  some  scale  which  may 

be  found,  LR  represents  coi2,  or  the  square  of  the  speed  HI  in 
revolutions  per  minute,  NR  represents  n22  and  MR  represents 
the  square  of  the  mean  speed  n  all  on  the  same  scale.  As  in 
engines  the  difference  between  n\  and  n2  is  never  large  it  is  fairly 
safe  to  assume  2n2  =  nz2  +  ni2  or  that  M  is  midway  between 
N  and  L. 

217.  Coefficient  of  Speed  Fluctuation.  —  Using  now  5  to  denote 


266  THE  THEORY  OF  MACHINES 

the  coefficient  of  speed  fluctuation,  then  5  is  denned  by  the 
relation 

_  n2  -  HI 
o  — 

n 

Now 

=  ^2  —  ni  =  nz  —  ni  _         2  —      2 

n  ~ 

Therefore 


or 

o 

25  =  r^—^ 


n* 

But  it  has  already  been  shown  that 

i  /     o       E\  2-jrni 

j/2coi2  =  -y-  =  tan  a\  and  since  coi  =  -^r-» 
J  i  OU 

therefore 


=  182.3  tan 


2X602 
or 

2  X  602 


Similarly,  n22  =  182.3  tan  «2;  thus  the  speeds  depend  on  a  only. 
Since  in  Fig.  156  the  base  OR  is  common  to  the  three  triangles 
with  vertices  at  N,  M  and  L,  it  follows  that 

RL  =  OR  tan  ai  =  OR  X 


and  RN  =  Cn22  where  C  =  7^-5  in  both  cases.     Further  gen- 
erally,  #M  =  Cn2. 


Then,  referring  to  the  formula  for  25,  which  is  25  =  -    ~T~ 
this  may  be  put  into  the  following  form : l 

RN  _  RL 

nS  -  nS        C          C        RN  -  RL       NL 

25  = 


n2  ^M  RM        ~  RM' 

/~v 

y 


Thus  NL  =  25  X  #M.     These  are  marked  in  Fig.  156. 

1  It  is  instructive  to  compare  this  investigation  with  the  corresponding  one 
for  governors  given  in  Sec.  183  and  Fig.  127 a. 


THE  PROPER  WEIGHT  OF  FLYWHEELS          267 

In  general,  a*  —  a\  is  a  small  angle  in  practice,  in  which  case 
M  may  be  assumed  midway  between  N  and  L  without  serious 
error,  and  on  this  assumption 

NM  =  ML  =  n2  X  6. 

The  foregoing  investigation  shows  that  the  shape  of  the  E  —  J 
diagram  has  a  very  important  effect  on  the  best  speed  for  a  given 
flywheel  and  the  best  weight  of  flywheel  at  the  given  speed. 
Thus,  Fig.  158  shows  one  form  of  this  curve  for  an  engine  to  be 
discussed  later,  while  Fig.  160  shows  two  other  forms  of  such 
curves  for  the  same  engine  but  different  conditions  of  loading. 
With  such  a  curve  as  that  on  the  right  of  Fig.  160,  the  best 
speed  condition  will  be  obtained  where  the  origin  0  is  located 
along  the  line  through  the  long  axis  of  the  figure.  In  order  to 
make  this  more  clear,  this  figure  is  reproduced  again  on  a  re- 
duced scale  at  Fig.  157  and  several  positions  of  the  origin  0  are 
drawn  in.  This  matter  will  now  be  discussed. 

218.  Effects  of  Speed  and  Flywheel  Weight. — Two  variables 
enter  into  the  problem,  namely  the  best  speed  and  the  most 
economical  weight  of  flywheel.  Now,  the  formula  connecting 
the  speed  with  the  angle  a  is  J^co2  =  tan  a,  Sec.  215,  so  that  the 
speed  depends  upon  the  angle  of  alone,  and  for  any  origin  along 
such  a  line  as  OF  there  is  the  same  mean  speed  since  a  is  constant 
for  this  line.  To  get  the  maximum  and  minimum  speeds  corre- 
sponding to  this  mean  speed,  tangents  are  drawn  from  0  to  the 
figure  giving  the  angles  ai  and  /*2  and  hence  coi  and  co2.  A  glance 
at  Fig.  157  shows  that  the  best  speed  corresponds  to  the  line  OF 
and  that  for  any  other  origin  such  as  Oi,  which  represents  a 
lower  mean  speed,  since  for  it  a  and  hence  tan  a  is  smaller,  there 
will  be  a  greater  difference  between  coi  and  co2  in  relation  to  co 
than  there  is  for  the  origin  at  0.  A  few  cases  have  been  drawn 
in,  and  it  is  seen  that  even  for  the  case  0±  which  represents  a 
higher  mean  speed  than  0  the  value  of  5  will  be  increased;  thus 
the  best  speed  corresponds  to  the  line  OF  and  its  value  is  found 
from  ^co2  =  tan  a. 

But  the  speed  variations  also  depend  on  the  weight  of  the 
flywheel  and  hence  upon  the  value  of^Tor  the  horizontal  distance 
of  the  origin  from  the  axis  Q'E'.  If  the  origin  was  at  0±,  there 
would  be  no  flywheel  at  all  but  the  speed  variation  taken  from  a 
scaled  drawing,  would  be  prohibitive  as  it  is  excessively  large. 
For  the  position  0  the  inertia  of  the  flywheel  is  represented  by 


268 


THE  THEORY  OF  MACHINES 


0  —  04  and  the  speed  variations  would  be  comparatively  small, 
but  if  the  origin  is  moved  up  along  OF  to  03,  the  speed  being  the 
same  as  at  0,  the  variations  will  be  increased  very  slightly,  but 
the  flywheel  weight  also  shows  a  greater  corresponding  decrease. 
Similarly,  Oi  corresponding  to  the  heaviest  wheel,  shows  a  varia- 
tion in  excess  of  0  and  nearly  equal  to  03,  and  02  with  the  same 


04 
FIG.  157. — Effect  of  speed  and  weight  of  flywheel. 


weight  of  wheel  as  at  0  shows  nearly  double  the  variation  that 
0  does. 

Thus,  increasing  the  weight  of  the  wheel  may  increase  the 
speed  variations  if  the  speed  is  not  the  best  one,  and  increas- 
ing the  speed  may  produce  the  same  result,  but  at  the  speed 
represented  by  OF,  the  heavier  the  wheel  the  smaller  will  be 
the  variation,  although  the  gain  in  steadiness  is  not  nearly 
balanced  by  the  extra  weight  of  the  wheel  beyond  a  certain 
point.  Frequently  the  operating  conditions  prevent  the  best 


THE  PROPER  WEIGHT  OF  FLYWHEELS 


269 


speed  being  selected,  and  if  this  is  so  it  is  clear  that  the  weight 
of  the  wheel  must  be  neither  too  large  nor  too  small. 

These  results  may  be  stated  as  follows:  For  a  given  machine 
and  method  of  loading  there  is  a  certain  readily  obtained  speed 
which  corresponds  to  minimum  speed  variations,  and  for  this 
best  value  the  variations  will  decrease  slowly  as  the  weight  of 
flywheel  is  increased.  For  a  certain  flywheel  weight  the  speed 
variations  will  increase  as  the  speed  changes  either  way  from  the 
best  speed,  and  an  increase  in  the  weight  of  the  flywheel  does 
not  mean  smaller  fluctuation  in  speed  unless  the  mean  speed 
is  suitable  to  this  condition. 

219.  Minimum  Mean  Speed.- — The  above  results  are  not  quite 
so  evident  nor  so  marked  in  a  curve  like  Fig.  158  but  the  same 


Plain  Line  is  for  Outward  Stroke 
Dotted  Line  is  for  Return  Stroke 


N~ 


17 


^ 


M 

i 
M& 


15 


20 


5  10 

FIG.  158. — Steam  engine  with  generator  or  turbine  pump  load. 

conditions  hold  in  this  case  also.  The  best  speed  is  much  more 
definitely  fixed  for  an  elongated  E  —  J  curve  and  becomes  less 
marked  as  the  boundary  of  the  curve  comes  most  nearly  to  the 
form  of  a  circle.  The  foregoing  investigation  further  shows  that 
no  point  of  the  E  —  J  curve  can  fall  below  the  axis  0  —  J,  because 
if  it  should  cut  this  axis,  the  machine  would  stop.  The  minimum 
mean  speed  at  which  the  machine  will  run  with  a  given  flywheel 
will  be  found  by  making  the  axis  0  —  J  touch  the  bottom  of  the 
curve,  and  finding  the  corresponding  mean  speed;  the  minimum 


270 


THE  THEORY  OF  MACHINES 


speed  will,  of  course,  be  zero,  since  a2  =  0.  The  minimum 
speed  of  operation  may  be  readily  computed  for  Figs.  158  and 
160  and  it  is  at  once  seen  that  the  right-hand  diagram  of  Fig.  160 
corresponds  to  a  larger  minimum  speed  than  any  of  the  others, 
that  is,  when  driving  the  air  compressor  the  engine  will  stop  at  a 
higher  mean  speed  than  when  driving  the  generator. 

220.  Numerical  Example  of  a  Steam  Engine. — The  principles 
already  explained  may  be  very  well  illustrated  in  the  case  of  the 
steam  engine  used  in  the  last  chapter,  which  had  a  cylinder  12>{6 
in.  diameter  and  30  in.  stroke  and  a  mean  speed  of  87  revolutions 
per  minute  for  which  co  =  9  radians  per  second.  The  form  of 
indicator  diagrams  and  loading  are  assumed  as  before  and  the 
engine  drives  a  turbine  pump  which  is  assumed  to  offer  constant 
resisting  torque.  The  weight  of  the  flywheel  is  required. 

Near  the  end  of  Chapter  XIII  is  a  table  containing  the  values 
of  J  and  5E  for  equal  parts  of  the  whole  revolution  and  for  con- 
venience these  results  are  set  down  in  the  table  given  herewith. 

TABLE  OF  VALUES  OF  J  and  E  for  12%  6  BY  30-iN.  ENGINE 


6,  degrees 

J,  total 

Jb  =  J  -  2,400 

8E,  foot-pounds 

0 

2,403.2 

3.2 

18 

2,405.4 

5.4 

-    233 

36 

2,410.9 

10.9 

+  1,387 

54 

2,416.8 

16.8 

+  1,414 

72 

2,420.5 

20.5 

+    699 

90 

2,420.7 

20.7 

+    186 

108 

2,418.5 

18.5 

191 

126 

2,412.7 

12.7 

461 

144 

2,407.9 

7.9 

-    646 

162 

2,404.4 

4.4 

-    858 

180 

2,403.2 

3.2 

-1,077 

198 

2,404.4 

4.4 

-    520 

216 

2,407.9 

7.9 

+    678 

234 

2,412.7 

12.7 

+  1,360 

252 

2,418.5 

18.5 

+    738 

270 

2,420.7 

20.7 

+    294 

288 

2,420.5 

20.5 

-      33 

306 

2,416.8 

16.8 

332 

324 

2,410.9 

10.9 

-    546 

342 

2,405.4 

5.4 

797 

360 

2,403.2 

3.2 

-1,058 

THE  PROPER  WEIGHT  OF  FLYWHEELS          271 

Selecting  the  axes  0'  — E'  and  0'  — J'&,  Fig.  158,  the  corresponding 
E-J  curve  is  readily  plotted  as  follows:  The  table  shows  that 
when  6  =  0°,  Jb  =  3.2  and  when  0  =  18°,  Jb  =  5.4,  the  gain  in 
energy  which  is  negative,  during  this  part  of  the  revolution  being 
6E  =  -  233  ft.-pds.  Starting  with  Jb  =  3.2  and  arbitrarily 
calling  E  at  this  point  1,000  ft.-pds.  gives  the  first  point  on  the 
diagram;  the  second  point  is  found  by  remembering  that  when 
Jb  has  reached  the  value  5.4  the  energy  has  decreased  by  233 
ft.-pds.,  so  that  the  point  is  located  on  the  line  Jb  =  5.4  and  233 
ft.-pds.  below  the  first  point.  The  third  point  is  at  Jb  =  10.9 
and  1,387  ft.-pds.  above  the  second  point  and  so  on. 

Now  draw  on  the  diagram  the  line  QM  to  represent  the  mean 
speed  co  =  9,  its  inclination  to  the  axis  of  Jb  being  a  where  tan  a  = 
J^co2  =  H  X  92  =  40.5.  The  actual  slope  on  the  paper  is  readily 
found  by  noticing  that  the  scales  are  so  chosen  that  the  same 
length  on  the  vertical  scale  stands  for  1,000  as  is  used  on  the  hori- 

1  000 

zontal  scale  to  represent  5,  the  ratio  being    '         =  200;  then  the 

5 

40  5 

actual  slope  of  QM  on  the  paper  is  — ^-  =  0.2025,  which  enables 

200 

the  line  to  be  drawn.  This  line  may  be  placed  quite  accurately 
by  making  the  perpendicular  distance  to  it  from  the  extreme 
lower  point  on  the  figure  equal  the  perpendicular  to  it  from  F 
(see  Figs.  156  and  158).  Thus  the  position  and  direction  of  the 
mean  speed  line  QM  are  known. 

Now  suppose  the  conditions  of  operation  require  that  the  max- 
imum speed  shall  be  1.6  per  cent,  above  the  minimum  speed, 
or  that  the  coefficient  of  speed  fluctuation  shall  be  1.6  per  cent. 

COo    —    6)1 

Then,  from  Sec.   217,   5  =  0.016,   that  is   5  =  -          -  =  0.016 

co 

and  the  problem  also  states  that  the  mean  speed  shall  be  co  =9 

=  -  — o .     On  comparing  these  two  results  it  is  found  that 

z 

co2  =    9.072  and  coi  =  8.928. 

On  substituting  these  two  values  in  the  equations  for  the 
angles,  the  results  are  tan  ai  =  J^coi2  =  H  X  79.709  =  39.854 
and  tan  «2  =  H^22  -  M  X  82.301  =  41.150  which  enables  the 
two  lines  HL  and  AF  to  be  drawn  tangent  to  the  figure  at  H 
and  F  and  at  angles  <*i  and  <*2  respectively  to  the  axis  of  Jb  (on 
the  paper  the  tangents  of  the  slopes  of  these  lines  will  be,  for 

AF  =  5  =  a2058  and  for  HL  =  =  0.1993).     These 


272  THE  THEORY  OF  MACHINES 

lines  are  so  nearly  parallel  that  their  distance  apart  vertically 
can  be  measured  anywhere  on  the  figure,  and  it  has  actually 
been  measured  along  NML}  the  distance  NL  representing 
3,180  ft.-pds. 

Referring  again  to  Fig.  156  it  is  seen  that  NR  =  OR  tan  «2 
and  LR  =  OR  tan  ai  and  by  combining  these  it  maybe  shown  that 

NL 

OR  =  -  —  .     Substituting  the  results  for  this  problem 

tan  «2  —  tan  on 

give 

O     I  QQ 

°R  --  41.160  '-39.854  =  2'453  -'•  +  *-/.+  ™- 

Hence,  the  moment  of  inertia  of  the  flywheel  should  be  2,430 
approximately,  which  gives  the  desired  solution  of  the  problem. 

221.  Method  of  Finding  Speed  Fluctuation  from  E-J  Dia- 
gram. —  The  converse  problem,  that  of  finding  the  speed  varia- 
tion corresponding  to  an  assumed  value  of  Ja,  has  been  solved  in 
the  previous  chapter  but  the  diagram  may  be  used  for  this  pur- 
pose also.  Thus,  let  co  =  9,  the  same  mean  '  speed  as  before, 

Tfl 

and  Ja  =  2,000.     Then,  since  J^co2  =  -j  =  tan  a  the  value  of 

J 

E  at  M  is  (2,000  +  25)  X  %  X  92  =  82,012.  The  points  N 
and  L  will  be  practically  unchanged  and  hence  at  N  the  value  of 
E2  is  82,012  +  J£(3,180)  =  83,602  ft.-pds.  and  the  value  of 

83  602 
co22  may  be  computed  from  the  relation  J^co22  =   0  '    ,   and  in  a 


similar  way  coi  may  be  found  and  the  corresponding  speed  varia- 


C02    — 

tion  5  = 


CO 

A  somewhat  simpler  method  may  be  used,  however,  by  refer- 
ring to  Fig.  156,  from  which  it  appears  that  NL  =  2n2<5.  Thus, 
2n2d  is  represented  by  3,180  ft.-pds.  and  n2  by  82,012  ft.-pds.,  from 
which  the  value  of  5  is  found  to  be  0.0175  which  corresponds  to 
a  speed  variation  of  1.75  per  cent. 

In  order  to  show  the  effect  of  making  various  changes,  let  the 
speed  of  the  engine  be  much  increased  to  say  136  revolutions  per 
minute  for  which  co  =  14.1,  and  let  the  speed  variation  be  still 
limited  to  1.6  per  cent.  The  line  QM  will  then  take  the  position 
Q'M  '  for  which  the  tangent  on  the  paper  is  %  and  the  distance 
corresponding  to  LN  measures  2,400  ft.-pds.  On  completing  the 
computations  the  moment  of  inertia  of  the  flywheel  is  found  to 
be  J0  =  740,  that  is  to  say  that  if  the  wheel  remains  of  the  same 


THE  PROPER  WEIGHT  OF  FLYWHEELS 


273 


diameter  it  need  be  less  than  one-third  of  the  weight  required 
for  the  speed  of  87  revolutions. 

The  diagram  Fig.  158  has  been  placed  on  the  correct  axis  and 
is  shown  in  Fig.  159  which  gives  an  idea  of  the  position  of  the 
origin  0  for  the  value  Ja  =  2,400  and  w  =  9. 


FIG.  159. 


222.  Effect  of  Form  of  Load  Curve  on  Weight  and  Speed.— To 
show  the  effects  of  the  form  of  load  curve  on  this  diagram  and 
on  the  speed  and  weight  of  the  flywheel,  the  curves  shown  in 
Fig.  160  have  been  drawn.  The  two  diagrams  shown  there  were 
made  for  the  same  engine  and  indicator  diagrams  as  were  used 
in  Fig.  158,  the  sole  difference  is  in  the  load  applied  to  the  engine. 
The  left-hand  diagram  corresponds  to  a  plunger  pump  connected 
in  tandem  with  the  steam  cylinder,  while  the  right-hand  diagram 
is  from  an  air  compressor  connected  in  tandem  with  the  steam 
cylinder.  The  effect  of  the  form  of  loading  alone  on  the  E  —  J 
diagram  is  most  marked  and  the  air  compressor  especially  pro- 
duces a  most  peculiar  result,  the  best  speed  here  being  definitely 
fixed  and  being  much  higher  than  for  either  of  the  other  cases, 
and  if  the  machine  is  run  at  this  speed  it  is  clear  that  the  weight 
of  the  flywheel  is  not  very  important  so  long  as  it  is  not  extremely 
small. 

It  is  needless  to  say  that  the  form  of  indicator  diagram  also 
produces  a  marked  effect  and  both  the  input  and  output  diagrams 
are  necessary  for  the  determination  of  the  flywheel  weight  and 
the  speed  of  the  machine.  The  curves  mentioned  are  sufficient 

18 


274 


THE  THEORY  OF  MACHINES 


to  show  that  the  weight  of  wheel  and  the  best  speed  of  operation 
depend  on  the  kind  of  engine  and  also  on  the  purpose  for  which 
it  is  used.  It  is  frequently  impossible,  practically,  to  run  an 
engine  at  the  speed  which  gives  greatest  steadiness  of  motion 
and  then  the  weight  of  wheel  must  be  selected  with  care  as  out- 
lined in  Sec.  218. 


7000 


20  ^ 


FIG.  160. — Left-hand  figure  is  for  a  plunger  pump  in  tandem  with  steam 
engine;  right-hand  figure  is  for  an  air  compressor  in  tandem  with  steam 
engine. 

223.  Numerical  Example  on  Four-cycle  Gas  Engine. — An  illus- 
tration of  the  application  to  a  gas  engine  of  the  four-cycle  type 
is  shown  at  Fig.  162,  this  being  taken  from  an  actual  case  of  an 
engine  direct-connected  to  an  electric  generator.  The  engine 
had  a  cylinder  14^  in.  diameter  and  22  in.  stroke  and  was  single- 
acting;  the  indicator  diagram  for  it  is  shown  at  Fig.  161.  The 
piston  and  other  reciprocating  parts  weighed  360  lb.,  while  the 
weight  of  the  connecting  rod  was  332  lb.,  and  its  radius  of  gyra- 
tion about  its  center  of  gravity  1.97  ft.,  the  latter  point  being 
24.3  in.  from  the  center  of  the  crankpin,  and  the  length  of  the 
rod  between  centers  was  55  in. 

There  were  two  flywheels  of  a  combined  weight  of  7,000  lb. 
and  the  combined  moment  of  inertia  of  these  and  of  the  rotor 
of  the  generator  was  1,600  (foot-pound  units). 


THE  PROPER  WEIGHT  OF  FLYWHEELS 


275 


The  form  of  the  E  —  J  diagram  for  this  case  is  given  in  Fig.  162 
and  differs  materially  in  appearance  from  any  of  those  yet  shown, 
and  the  best  speed  is  much  more  difficult  to  determine  because 
of  the  shape  of  the  diagram.  The  actual  speed  of  the  engine 


400  r 


200 


ICO 


o1- 


FOUR-CYCLE  GAS  ENGINE 
DIAGRAM 


Zero  JLine 

FIG.  161. 


'2~~~    4          6          8        10       12        14        16        18  J $ 
FIG.  162. — Gas  engine  driving  dynamo. 

was  172  revolutions  per  minute  and  for  this  value  the  sloping 
lines  on  the  diagram  have  been  drawn.  The  mean-speed  line 
would  have  an  inclination  to  the  axis  of  Jb  given  by  tan  a  = 

Ifa2  =  162  and  its  slope  on  the  paper  would  be  1  gr.A  of  this, 

1,DUU 


276  THE  THEORY  OF  MACHINES 

since  the  vertical  scale  is  1,500  times  the  horizontal;  thus  the 

162 

tangent  of  the  actual  slope  is  1          =  0.108  and  the  lines  are 

l,oUu 

drawn  with  this  inclination. 

The  total  height  of  the  diagram  is  31,200  ft.-pds.  and  using  the 
value  Ja  =  1,600,  the  mean  value  of  E  is  162  X  1,600  =  259,200 
ft.-pds.  so  that  the  speed  variation  is 

31,200 
5  =  Ji  X         2QQ  =  0.0602  or  6.02  per  cent. 


The  engine  here  described  was  installed  to  produce  electric 
light  and  it  is  perfectly  evident  that  it  was  entirely  unsuited  to 
its  purpose  as  such  a  large  speed  variation  is  quite  inadmissible. 
Owing  to  the  peculiar  shape  of  this  diagram  and  the  fact  that  the 
tangent  points  touch  it  on  the  left-hand  side,  it  appears  that  the 
distance  between  them  will  not  be  materially  changed  by  any 
reasonable  change  of  slope  of  the  lines,  so  that  if  the  speed  re- 
mains constant  at  172  revolutions  per  minute  the  value  of  Ja  or 
the  flywheel  weight  is  inversely  proportional  to  the  speed  varia- 
tion and  flywheels  of  double  the  weight  would  reduce  the  fluc- 
tuation to  about  3  per  cent. 

A  change  in  speed  would  bring  an  improvement  in  conditions 
and  the  results  may  readily  be  worked  out. 

QUESTIONS  ON  CHAPTER  XIV 

1.  Show  the  effect  of  the  following:  (a)  increase  in  flywheel  weight,  con- 
stant speed;  (b)  decrease  under  same  conditions;  (c)  increase  in  speed  with 
same  flywheel;  (d)  increase  and  decrease  in  both  weight  and  speed;  all  with 
reference  to  a  gas  engine. 

2.  What  would  be  the  shape  of  the  E-J  diagram  for  a  geared  punch, 
neglecting  the  effect  of  the  reciprocating  head? 

3.  If  the  connecting  rod  and  piston  of  an  engine  are  neglected,  what  would 
be  the  shape  of  the  E-J  curve?     What  would  be  its  dimensions  in  the  two 
examples  of  the  chapter? 

4.  What  would  be  the  best  speed  for  the  steam  engine  given  in  the  text 
when  driving  the  three  different  machines?     At  what  mean  speed  would  the 
engine  stop  in  the  three  cases? 

6.  What  would  be  the  best  speed  for  the  gas  engine  and  at  what  mean 
speed  would  it  stop? 

6.  What  flywheel  weight  would  reduce  the  speed  variation  5  per  cent,  for 
the  steam  engine? 

7.  Examine  the  effect  on  the  E-J  diagram  for  the  engine-driven  com- 
pressor if  a  crank  for  operating  the  latter  is  set  at  90°  to  the  engine  crank. 
What  effect  has  this  on  the  best  speed? 


CHAPTER  XV 

ACCELERATIONS    IN    MACHINERY    AND    DISTURBING 
FORCES  DUE  TO  THE  INERTIA  OF  THE  PARTS 

224.  General  Effects  of  Accelerations. — It  has  become  a  prac- 
tice in  modern  machinery  to  operate  it  at  as  high  a  speed  as  possi- 
ble in  order  to  increase  its  output.  Where  the  machines  con- 
tain parts  that  are  not  moving  at  a  uniform  speed,  such  as  the 
connecting  rod  of  an  engine  or  the  swinging  jaw  of  a  rock  crusher, 
the  variable  nature  of  the  motion  requires  alternate  acceleration 
and  retardation  of  these  parts,  to  produce  which  forces  are  re- 
quired. These  alternate  accelerations  and  retardations  cause 
vibrations  in  the  machine  and  disturb  its  equilibrium;  almost 
everyone  is  familiar  with  the  vibrations  in  a  motor  boat  with  a 
single-cylinder  engine,  and  many  law-suits  have  resulted  from 
the  vibrations  in  buildings  caused  by  machinery  in  shops  and 
factories  nearby. 

These  vibrations  are  very  largely  due  to  the  irregular  motions 
of  the  parts  and  to  the  accelerating  forces  due  to  this,  and  the 
forces  increase  much  more  rapidly  than  the  speed,  so  that  with 
high-speed  machinery  the  determination  of  these  forces  becomes 
of  prime  importance,  and  they  are,  indeed,  also  to  be  reckoned 
with  in  slow-speed  machinery,  as  there  are  not  a  few  cases  of 
slow-running  machines  where  the  accelerating  forces  have  caused 
such  disturbances  as  to  prevent  the  owners  operating  them. 

Again,  in  prime  movers  such  as  reciprocating  engines  of  all 
classes,  the  effective  turning  moment  on  the  crankshaft  is  much 
modified  by  the  forces  necessary  to  accelerate  the  parts;  in  some 
cases  these  forces  are  so  great  that  the  fluid  pressure  in  the  cylin- 
der will  not  overcome  them  and  the  flywheel  has  to  be  drawn 
upon  for  assistance.  The  troubles  are  particularly  aggravated 
in  engines  of  high  rotative  speed  and  appear  in  a  most  marked 
way  in  the  high-speed  steam  engine  and  in  the  gasoline  engines 
used  in  automobiles. 

The  forces  required  to  accelerate  the  valves  of  automobile 
engines  may  also  be  so  great  that  the  valve  will  not  always  re- 
main in  contact  with  its  cam  but  will  alternately  leave  it  and  re- 
turn again,  thus  causing  very  noisy  and  unsatisfactory  operation. 

277 


278 


THE  THEORY  OF  MACHINES 


Specific  problems  involving  the  considerations  outlined  above 
will  be  dealt  with  later  but  before  such  problems  can  be  solved 
it  will  be  necessary  to  devise  a  means  of  finding  the  accelerations 
of  the  parts  in  as  convenient  and  simple  a  way  as  possible,  and 
this  will  now  be  discussed. 

225.  The  Acceleration  of  Bodies. — The  general  problem  of  accel- 
eration in  space  has  not  much  application  in  machinery,  so  that  the 
investigation  will  here  be  confined  to  a  body  moving  in  one  plane, 
which  will  cover  most  practical  cases.     Let  a  body  having  weight 

w  Ib.  move  in  a  plane  at  any  instant;  its  mass  will  be  m  =  — •  and 

y 

by  the  principle  of  the  virtual  center  as  outlined  in  Chapter  III, 
its  motion  is  equivalent  at  any  instant  to  one  of  rotation  about 
a  point  in  the  plane  of  motion,  which  point  may  be  near  or  remote 
according  to  the  nature  of  the  motion;  if  the  point  is  infinitely 
distant  the  body  moves  in  a  straight  line,  or  has  a  motion  of 
translation. 

226.  Normal  and  Tangential  Acceleration. — Let  Fig.  163  repre- 

sent a  body  moving  in  the  plane  of 
the  paper  and  let  0  be  its  virtual 
center  relative  to  the  paper,  0  being 
thus  the  point  about  which  the  body 
is  turning  at  the  instant.  The  body 
is  also  assumed  to  be  turning  with 
variable  speed,  but  at  the  instant  when 
it  is  passing  through  the  position 
shown  let  its  angular  velocity  be  w 
radians  per  second.  Any  point  P  in 
the  body  will  travel  in  a  direction 
normal  to  OP,  Sec.  34,  in  the  sense 
indicated,  as  this  corresponds  with 
the  sense  of  the  angular  velocity. 
This  point  P  has  accelerations  in  two 
directions :  (a)  Since  the  body  is  mov- 
ing about  0  at  variable  angular  velocity  it  will  have  an  acceler- 
ation in  the  direction  of  its  motion,  that  is,  normal  to  OP]  and 
(6)  it  will  have  an  acceleration  toward  0  even  if  o>  is  constant, 
since  the  point  is  being  forced  to  move  in  a  circle  instead  of  a 
straight  line.  The  first  of  these  may  be  called  the  tangential 
acceleration  of  the  point  since  it  is  the  acceleration  of  the  point 
along  a  tangent  to  its  path,  while  the  second  is  its  normal  acceler- 


ACCELERATIONS  IN  MACHINERY  279 

ation  for  similar  reasons.  Every  point  in  a  body  rotating  at  a 
given  instant  has  normal  acceleration,  no  matter  what  kind  of 
motion  the  body  has,  but  it  will  only  have  tangential  acceleration 
if  the  angular  velocity  of  the  body  is  variable.  If  the  body  has 
a  motion  of  translation  it  can  only  have  tangential  acceleration. 
In  Fig.  164  let  OP  be  drawn  separately,  its  length  being  r  ft. 
and  at  the  time  8t  sec.  later  let  OP  be  in  the  position  OQ,  the  angle 
QOP  being  dd,  so  that  the  body  has  turned  through  the  angle  50 
radians  in  dt  sec.  The  angular  velocity  when  in  the  position 
OP  is  co  radians  per  second  and  in  the  position  OQ  is  assumed  to 
be  co  +  5co  radians  per  second;  thus  the  gain  in  angular  velocity 
is  5 co  radians  per  second  in  the  time  dt,  or  the  angular  acceleration 

of  the  body  is  a  =  —  radians  per  second  per  second.     Now 

Ov 

draw  the  corresponding  velocity  triangle  as  shown  on  the  right 


FIG.  164. 

of  Fig.  164,  making  SM  normal  to  OP,  equal  to  OP  X  o>  =  rco 
ft.  per  second  and  SN  normal  to  OQ  equal  to  OP(co  +  5co)  = 
r(«  +  5co)  ft.  per  second,  so  that  the  gain  in  linear  velocity  in  the 
time  dt  sec.  is  MN  ft.  per  second  and  its  components  in  the  nor- 
mal and  tangential  directions  are  MR  and  RN  ft.  per  second 
respectively. 

The  normal  gain  in  velocity  of  the  point  P  in  the  time  8t  is 
MR,  so  that  its  normal  acceleration  is 

MR      rco50  50 

"N  =  -jr  —  —^7-  =  rco—  =  rco2  ft.  per  second  per  second, 
ot  ot  ot 


and  similarly  the  tangential  acceleration  is 

pr  =  **  = 

U 

per  second. 


NR      SN-SR      r(co  +  5co)  -  rco        5co 

~  ~  =T=roL  ft' per  second 


280  THE  THEORY  OF  MACHINES 

The  sense  of  PT  is  determined  by  that  of  a  while  PN  is  always 
radially  inward  toward  the  center  0.  Thus,  the  normal  accelera- 
tion of  the  point  is  simply  its  instantaneous  radius  of  rotation 
multiplied  by  the  square  of  the  angular  velocity  of  its  link  while 
the  tangential  acceleration  of  the  point  is  the  radius  of  rotation 
of  the  point  multiplied  by  the  angular  acceleration  of  its  link. 
Where  the  link  turns  with  uniform  velocity  a  =  0  and  therefore 
PT  =  0,  but  PN  can  only  be  zero  if  to  is  zero,  which  means  that  the 
link  is  at  rest  or  has  a  motion  of  translation.  In  the  latter  case 
the  link  can  only  have  tangential  acceleration. 

227.  Graphical  Construction. — Returning  now  to  Fig.  163,  the 
normal  acceleration  of  P  or  PN  is  rco2  toward  0,  then  take  the 
length  OP  to  represent  this  quantity,  thus  adopting  the  scale  of 
—  a?2 :  1 ;  this  is  negative  since  the  line  OP  represents  the  accelera- 
tion rco2  in  the  direction  and  sense  PO.     Then  the  tangential 
acceleration  PT  will  be  represented  by  a  line  normal  to  OP,  its 

TCt 

length  will  be  — — -2  since  the  scale  is  —  co2 : 1,  and  its  sense  is  to  the 

Tct 

right,  since  the  scale  is  negative,  hence  draw  PP"  =  -  — ^     Now  if 

0  be  joined  to  P"  then  OP"  =  vector  sum  OP  +  PP"  or  OP"  = 
PN  +  PT  which  will  therefore  represent  the  total  acceleration 
of  P,  that  is  the  total  acceleration  of  P  is  P"0  X  <o2  in  the  direc- 
tion and  sense  P"0.  It  may  very  easily  be  shown  that  in  order 
to  find  the  acceleration  of  any  other  point  R  on  this  body  at  the 
given  instant  it  will  only  be  necessary  to  locate  a  point  R"  bear- 
ing the  same  relation  to  OP"  that  R  does  to  OP,  the  acceleration 
of  R,  which  is  represented  by  OR" ',  being  #"0.co2  and  its  direction 
and  sense  R"0.  The  acceleration  of  R  with  reference  to  P  is 
R"P"u*. 

228.  Application  to  Machines. — The  accelerations  may  now 
be  found  for  machines  and  the  first  case  considered  will  be  as 
general  as  possible,  the  machine  being  one  of  four  links  with 
four  turning  pairs,  Fig.  165.     Let  the  angular  velocity  co  and  the 
angular  acceleration  a  of  the  selected  primary  link  a  be  known,  it 
is  required  to  find  the  angular  accelerations  of  the  other  links  as 
well  as  the  linear  accelerations  of  different  points  in  them.     From 
the  phorograph,  Chapter  IV,  the  angular  velocities  of  the  links 

b  and  c  are  o>&  =  7-00  and  coc  =  —  to  respectively,  and  these  may 

U  C 

readily  be  found.     Further,  if  05  and  ac  represent  the,  as  yet 


ACCELERATIONS  IN  MACHINERY  281 

unknown,  angular  accelerations  in  space  of  b  and  c  respectively, 
and  also  if  QN  and  QT  represent  respectively  the  normal  and 
tangential  accelerations  of  Q  with  regard  to  P,  the  point  about 
which  b  is  turning  relative  to  a,  and  RN  and  RT  have  the  same 
significance  as  regards  R  relative  to  Q,  then  the  previous  para- 
graph enables  the  following  relations  to  be  established: 

PAT  =  aco2;  PT  =  act]  QN  =  frcob2;  QT  =  bab',  RN  =  c.coc2  and  RT 

=    COic. 

Using  the  principle  of  vector  addition  the  total  acceleration  of 
R  with  regard  to  0  is  the  vector  sum  of  the  accelerations  of  R 


FIG.  165. 

with  regard  to  Q,  of  Q  with  regard  to  P  and  of  P  with  regard  to  0. 
But  as  R  and  0  are  stationary,  the  total  acceleration  of  R  with 
regard  to  0  is  zero,  hence,  the  sum  of  the  above  three  accelera- 
tions is  zero,  or 

RT  +  RN  +  QT  +  QN  +  PT  +  PN  =  0, 

that  is,  the  vector  polygon  made  up  with  these  accelerations  as 
its  sides  must  close,  or  if  the  polygon  be  started  at  0  it  will  end 
at  0  also. 

Now  the  point  P"  may  be  located  according  to  the  method 
previously  given,  and  in  order  to  locate  Q",  giving  the  total 
acceleration  of  Q,  proceed  from  P"  to  0  by  means  of  the  vectors 
QN  +  QT  +  RN  +  RT.  The  direction  and  sense  of  both  Qv 
and  RN  are  known,  they  are  respectively  QP  and  RQ,  further,  the 
direction,  but  not  the  sense  of  QT  and  of  RT  is  known,  in  each  case 


282  THE  THEORY  OF  MACHINES 

it  is  normal  to  the  link  itself,  or  QT  is  normal  to  b  and  RT  is  normal 
to  c. 

In  order  to  represent  the  results  graphically  they  may  be  put 
into  the  following  convenient  form : 

QN  =  feW62   =  &>2     :       -  X  a,2 


and 

2 


v/       .2 

~   A   to 
C 

and  remembering  that  the  scale  for  the  diagram  is  —  w2:!,  draw 

QN      b'2  RN      c'2 

P" A  =  — f  =  -j-  and  follow  it  with  AB  =  — *  =  — ,  the  negative 


sign  having  been  taken  into  account  by  the  sense  in  which  these 
are  drawn.  The  polygon  from  B  to  0  may  now  be  completed 
by  adding  the  vectors  QT  and  RT,  and  as  the  directions  of  these 
are  known,  the  process  is  evidently  to  draw  from  0  the  line  OC 
in  the  direction  RT,  that  is  normal  to  c,  and  from  B  the  line  BC 
normal  to  6,  which  is  in  the  direction  of  QT,  these  lines  inter- 
secting at  the  point  C.  Then  it  is  evident  that  BC  represents 
QT  on  the  scale  —  co2  to  1,  and  that  OC  represents  RT  on  the  same 
scale,  so  that  in  the  diagram  OPP"AECQ"0  it  follows  that 
OP  =  PNj  PP"  =  PT,  P"A  =  QN,  AB  =  RN,  BC  =  QT  and  CO 
=  RT,  all  on  the  scale  —  co2  to  1.  Complete  the  parallelogram 
CBAQ";  then  OP"  =  PN  +  PT,  P"Q"  =  Q»  +  QT  and  Q"0  = 
RN  +  RT,  and  therefore,  the  vector  triangle  OP"Q!'R"  gives  the 
vector  acceleration  diagram  of  all  the  links  on  the  machine. 

229.  Acceleration  of  Points. — The  linear  acceleration  of  any 
point  such  as  G  on  b  is  readily  shown  to  be  represented  by  OG" 
and  to  be  equal  to  G"0.u2,  where  the  point  G"  divides  P"Q"  in 
the  same  way  that  G  divides  PQ,  the  direction  and  sense  of  the 
acceleration  of  G  is  G"0.  Similarly,  the  acceleration  of  H  in 
c  is  H"0.u2  in  magnitude,  direction  and  sense  where  H"  divides 
OQ"(R"Q")  in  the  same  way  as  H  divides  RQ.  In  this  way  the 
linear  acceleration  of  any  point  on  a  machine  may  be  directly 
determined. 

Angular  Accelerations  of  the  Links. — The  angular  accelera- 
tions of  the  links  may  be  found  as  follows.  Since  QT  =  AQ"  X  —  co2 

=  bab,  then  bab  =  -  AQ".u*  or  -  ab  =  AQ"  X  ^-  so  that  the 

o 

length  AQ"  represents  ab,  the  angular  acceleration  of  the  link 


ACCELERATIONS  IN  MACHINERY  283 

6,  and  similarly  CO  represents  the  angular  acceleration  <xc  of 

c  or  <xc  =  —  CO  X  — •     The  sense  of  these  angular  accelerations 
c 

may  be  found  by  noticing  the  way  one  turns  to  them  in  going 
from  the  corresponding  normal  acceleration  line;  thus,  in  going 
from  PN  to  PT  one  turns  to  the  right,  in  going  from  QN(P"A)  to 
QT(AQ")  the  turn  is  to  the  left  and  hence  o:fe  is  in  opposite  sense 
to  a,  and  by  a  similar  process  of  reasoning  ac  is  in  the  same  sense 
as  a.  Thus,  in  the  position  shown  in  the  diagram,  Fig.  165,  the 
angular  velocities  are  increasing  for  the  links  a  and  c,  and  that 
of  the  link  b  is  also  increasing  since  both  a&  and  w&  are  in  opposite 
sense  to  a  and  co. 

It  will  be  found  that  the  method  described  may  be  applied  to 
any  machine  no  matter  how  complicated,  and  with  comparative 
ease.  The  construction  resembles  the  phorograph  of  Chapter 
IV,  which  it  employs,  and  hence  this  latter  chapter  must  be  care- 

6'2 
fully  read.     Simple  graphical  methods  for  finding  -=--,  etc.,  may 

be  made  up,  one  of  which  is  shown  in  the  applications  given 
hereafter. 

THE  FORCES  DUE  TO  ACCELERATIONS  OF  THE  MACHINE  PARTS 

230.  The  real  object  of  determining  the  accelerations  of  points 
and  links  in  a  machine  is  for  the  purpose  of  finding  the  forces 
which  must  be  applied  on  the  machine  parts  in  order  to  produce 
these  accelerations  and  also  to  learn  the  disturbing  effects  pro- 
duced in  the  machine  if  the  accelerations  of  the  parts  are  not 
balanced  in  some  way.  The  investigation  of  these  disturbing 
effects  will  now  be  undertaken,  the  first  matter  dealt  with  being 
the  forces  which  must  be  applied  to  the  links  to  produce  the 
changing  velocities. 

It  is  shown  in  books  on  dynamics,  that  if  a  body  having  plane 

motion,  has  a  weight  w  Ib.  or  mass  m  =  —  and  an  acceleration  of 

g 

its  center  of  gravity  of  /  ft.  per  second  per  second,  then  the  force 
necessary  to  produce  this  acceleration  is  mf  pds.,  and  this  force 
must  act  through  the  center  of  gravity  and  in  the  direction  of  the 
acceleration  /.  In  many  cases  the  body  also  rotates  with  variable 
angular  velocity,  or  with  angular  acceleration,  in  which  case  a 
torque  must  act  on  the  body  in  any  position  to  produce  this 
variable  rotary  motion,  and  if  the  body  has  a  moment  of  inertia 


284 


THE  THEORY  OF  MACHINES 


I  about  its  center  of  gravity  and  angular  acceleration  a  radians 
per  second  per  second  this  torque  must  have  a  magnitude  of 
I  X  a  ft.-pds.  Let  the  mass  of  the  link  be  so  distributed  that  its 
radius  of  gyration  about  the  center  of  gravity  is  k;  then  I  = 
mk2  and  the  torque  is  mk2a.  For  proof  of  this  the  reader  is 
referred  to  books  on  dynamics. 

To  take  a  specific  case  let  a  machine  with  four  links  be  selected, 
as  illustrated  in  Fig.  166,  and  let  the  vector  acceleration  diagram 


FIG.  166. — Disturbing  forces  due  to  mass  of  rod. 

OP"Q"0,  as  well  as  the  phorograph  OP'Q'O  be  found,  as  already 
explained;  it  is  required  to  find  the  force  which  must  be  exerted 
on  any  link  such  as  b  to  produce  the  motion  which  it  has  in  the 
given  position.  Let  G  be  the  center  of  gravity  of  the  link  and 
let  its  weight  be  Wb  lb.  and  its  moment  of  inertia  about  G  be 
represented  by  Ib  in  feet  and  pound  units;  then  7&  =  mbkb2  where 

mb  =  —  and  kb  is  the  radius  of  gyration  about  the  point  G.     From 

i/ 

the  vector  diagram  it  is  assumed  that  the  angular  acceleration 
otb  has  been  found;  also  the  acceleration  of  G,  which  is  G"0  X  co2. 
To  produce  the  acceleration  of  G  a  force  must  act  through  it 
of  amount  F  =  m  X  G"0  X  co2  in  the  direction  and  sense  G"0, 
while  to  produce  the  angular  acceleration  a  torque  T  must  act 
on  the  link  of  amount  T  =  /&o&  =  m^k^a*.  The  torque  T  may 


ACCELERATIONS  IN  MACHINERY  285 

be  produced  by  a  couple  consisting  of  two  parallel  forces  ac-ting 
in  opposite  sense  and  at  proper  distance  apart,  and  these  forces 
may  have  any  desired  magnitude  so  long  as  their  distance  apart 
is  adjusted  to  suit.  For  convenience  let  each  of  the  forces  be 
selected  equal  to  F;  then  the  distance  x  ft.  between  them  will 
be  found  from  the  relation  T  =  Fx. 

Now,  as  this  couple  may  act  in  any  position  on  the  link  b  let 
it  be  so  placed  that  one  of  the  forces  passes  through  G  and  the 
two  forces  have  the  same  direction  as  the  acceleration  of  G. 
Further,  let  the  force  passing  through  G  be  the  one  which  acts 
in  opposite  sense  to  the  accelerating  force  F',  this  is  shown  on 
Fig.  166.  Now  the  accelerating  force  F  and  one  of  the  forces 
F  composing  the  couple  act  through  G  and  balance  one  another 
and  thus  the  accelerating  force  and  the  couple  producing  the 
torque  reduce  to  a  single  force  F  whose  magnitude  is  ra&.G"0.a>2, 
whose  direction  and  sense  are  the  same  as  the  acceleration  of  the 
center  of  gravity  G  of  b,  and  which  acts  at  a  distance  x  from  G, 
determined  by  the  relation  T  =  Fx}  and  on  that  side  of  G  which 
makes  the  torque  act  in  the  same  sense  as  the  angular  accelera- 
tion a&. 

The  distance  x  of  the  force  F  from  G  may  be  found  as  follows  : 

2 

Since  QT  =  bab  =  Q"A  X  co2,  Fig.  165,  then  ab  =  Q"A  X  ~,  because 

o 

the  line  AQ"  represents  QT  on  a  scale  —  co2  :  1. 

Also  T  =  Ibab  =  mbkb  2  X^V-  X  co2 

b 

and  F  =  mb.G"0.u\ 

Q"A 


T  —  _ 

therefore  *-  —    -^r^       —  •  -^-where  -^-  is  a 


Q"A 
constant,  so  that  x  =  const.  X  ^7777  which  ratio  can  readily  be 

Or    U 

found  for  any  position  of  the  mechanism.  This  gives  the  line  of 
action  of  the  single  force  F  and,  having  found  the  position  of  the 
force,  let  M  be  its  point  of  intersection  with  the  axis  of  link  b. 
Now  find  M'  the  image  of  M  and  move  the  force  from  M  to  its 
image  M']  then  the  turning  moment  necessary  on  the  link  a  to  ac- 
celerate the  link  b  is  Fh,  where  h  is  the  shortest  distance  from 
0  to  the  direction  of  F,  Fig.  166. 


286 


THE  THEORY  OF  MACHINES 


This  completes  the  problem,  giving  the  force  acting  on  the  link 
and  also  the  turning  moment  at  the  link  a  necessary  to  produce 
this  force.  The  same  construction  may  be  applied  to  each  of 
the  other  links  such  as  a  and  c  and  thus  the  turning  moment  on  a 
necessary  to  accelerate  the  links  may  be  found  as  well  as  the 
necessary  force  on  each  link  itself. 

DETERMINATION  OF  THE  STRESSES  IN  THE  PARTS  DUE  TO 
THEIR  INERTIA 

231.  The  results  just  obtained  may  be  used  to  find  the  bending 
moment  produced  in  any  link  at  any  instant  due  to  its  inertia. 


\^*~ 

^ 

^~-- 

-^ 

.... 


,      dm 

, 

b      c 

1 

3       ( 

;             ( 

'R 
FIG.  167. — Bending  forces  on  rod  due  to  its  inertia. 

Any  part  such  as  the  connecting  rod  of  an  engine  is  subject  to 
stresses  due  to  the  transmission  of  the  pressure  from  the  piston 
to  the  crankpin,  but  in  addition  to  this  the  rod  is  continually 
being  accelerated  and  retarded,  these  changes  of  velocity  pro- 
ducing bending  stresses  in  the  rod,  and  these  latter  stresses  may 
now  be  determined. 

To  make  the  case  as  general  as  possible,  let  OPQR,  Fig.  167, 
represent  a  machine  for  which  the  vector  acceleration  diagram  is 
OP"Q"0,  it  is  required  to  find  the  bending  moment  in  the  rod  b 
due  to  its  inertia.  Lay  off  at  each  point  on  b  the  acceleration  of 
that  point;  thus  make  PA\,  GCi,  QBi,  etc.,  equal  and  parallel  re- 
spectively to  OP",  OG",  QO",  etc.,  obtaining  in  this  way  the 
curve  AiCiBi. 

Now  resolve  the  accelerations  at  each  point  in  b  into  two  parts, 


ACCELERATIONS  IN  MACHINERY  287 

one  normal  to  b  and  the  other  parallel  to  the  link.  Thus  PA  is 
the  acceleration  of  P  normal  to  6,  and  GC  and  QB  are  the  corre- 
sponding accelerations  for  the  points  G  and  Q  respectively.  In 
this  way  a  second  curve  ACB  may  be  drawn,  and  the  perpen- 
dicular to  b  drawn  from  any  point  in  it  to  the  line  ACB  represents 
the  acceleration  at  the  given  point  in  b  in  the  direction  normal 
to  the  axis  of  the  latter,  the  scale  in  all  cases  being  —  co2  : 1. 
Thus  the  acceleration  of  P  normal  to  b  is  AP.w2,  and  so  for  other 
points. 

Now  let  the  rod  be  placed  as  shown  on  the  right-hand  side  of 
Fig.  167  with  the  acceleration  curve  ACB  above  it  to  scale. 
Imagine  the  rod  divided  up  into  equal  short  lengths  one  of  which 

is  shown  at  D,  having  a  weight  5w  Ib.  and  mass  dm  =  — ,  and 

let  the  normal  acceleration  at  this  point  be  represented  by  DE. 
Should  the  rod  be  of  uniform  section  throughout  its  length  all 
the  small  masses  like  dm  will  be  equal  since  all  will  be  of  the  same 
weight  dw,  but  if  the  rod  is  larger  at  the  left-hand  end  than  at 
the  right-hand  end,  then  the  values  of  dm  will  decrease  in  going 
along  from  P  to  Q.  Now  tlie  force  due  to  the  acceleration  of 
the  small  mass  is  equal  to  dm  multiplied  by  the  acceleration 
corresponding  to  DE  and  this  force  may  be  set  off  along  DE 
above  D.  Proceeding  in  this  way  for  the  entire  length  of  the 
rod  gives  the  dotted  curve  as  shown  which  may  be  looked  upon  as 
the  load  curve  for  the  rod  due  to  its  acceleration.  From  this 
load  curve  the  bending  moments  and  stresses  in  the  rod  may  be 
determined  by  the  well-known  methods  used  in  statics. 

For  a  rod  of  uniform  cross-section  throughout  the  acceleration 
curve  ACB  will  also  be  a  load  curve  to  a  properly  selected  scale, 
but  with  the  ordinary  rods  of  varying  section  the  work  is  rather 
longer.  In  carrying  it  out,  the  designer  usually  soon  finds  out 
by  experience  the  position  of  the  mechanism  which  corresponds 
to  the  highest  position  of  the  acceleration  curve  ACB,  and  the 
accelerations  being  the  maximum  for  this  position  the  rod  is 
designed  to  suit  them.  A  very  few  trials  enable  this  position 
to  be  quickly  found  for  any  mechanism  with  which  one  is  not 
familiar. 

The  process  must,  of  course,  be  carried  out  on  the  drafting 
board. 

232.  To  Find  the  Accelerations  of  the  Various  Parts  of  a  Rock 
Crusher. — In  order  to  get  a  clearer  grasp  of  the  principles  in- 


288 


THE  THEORY  OF  MACHINES 


volved,  a  few  applications  will  be  made,  the  first  case  being  that 
of  the  rock  crusher  shown  in  Fig.  168.  The  mechanism  of  the 
crusher  is  shown  on  the  left  and  has  not  been  drawn  closely  to 
scale  as  the  construction  is  more  clear  lor  the  proportions  shown. 
A  crank  OP  is  driven  at  uniform  speed  by  a  belt  pulley  on  the 
shaft  0  and  to  this  crank  is  attached  the  long  connecting  rod  PQ. 
The  swinging  jaw  of  the  crusher  is  pivoted  to  the  frame  at  T 
and  connected  to  PQ  by  the  rod  SQ,  while  another  rod  QR  is 
pivoted  to  the  frame  at  R.  As  OP  revolves  Q  swings  in  an  arc 
of  a  circle  about  R}  giving  the  jaw  a  swinging  motion  about  T 


Q" 


FIG.  168. — Rock  crusher. 

and  crushing  between  the  jaw  and  the  frame  any  rocks  falling 
into  the  space.  In  large  crushers  the  jaw  is  very  heavy  and  its 
variable  velocity,  or  acceleration,  sometimes  sets  up  very  serious 
vibrations  in  buildings  in  which  it  is  placed.1 

The  acceleration  diagram  is  shown  on  the  right  and  there  is 
also  drawn  the  upper  end  of  the  rod  b  and  the  whole  of  the  crank 
a.  It  is  to  be  noted  that  the  actual  mechanism  may  be  drawn 
to  as  small  a  scale  as  desired  and  the  diagram  to  the  right  to  as 
large  a  scale  as  is  necessary,  because  in  the  phorograph  and  the 
acceleration  diagram  only  the  directions  of  the  links  are  required 
and  these  may  be  easily  obtained  from  the  small  scale  drawing 

1  See  article  by  PROF.  O.  P.  HOOD  in  American  Machinist,  Nov.  26,  1908. 


ACCELERATIONS  IN  MACHINERY  289 

shown.  The  phorograph  of  the  mechanism  and  acceleration  dia- 
gram should  give  no  difficulty  because  the  mechanism  is  simply 
a  combination  of  two  four  link  mechanisms,  OPQR  and  RQST 
exactly  similar  to  that  shown  in  Fig.  165  and  already  dealt  with. 
The  crank  OP  has  been  chosen  as  the  primary  link. 

The  crank  OP  is  assumed  to  turn  at  uniform  speed  of  co  radians 
per  second.  For  the  phorograph,  OQ'  parallel  to  RQ  meeting  6 
produced  gives  Q'  and  P'Q'  =  V  and  OQ'  =  c';  further  OS'  par- 
allel to  ST  meeting  Q'S'  parallel  to  QS  gives  S'  and  Q'S'  =  e' 
while  S'O  gives  /'.  The  points  R'  and  Tr  lie  at  0. 

For  the  acceleration  diagram  P"  lies  at  P  since  a  is  assumed  to 
run  at  uniform  speed;  then,  following  the  method  already  de- 

o 

scribed  in  Sec.  228,  lay  off  P"A  =  -j-  and  AB  parallel  to  c  and 

c/2 

of  length  AB  =  — ,  and  finish  the  diagram  by  making  BC  per- 
c 

pendicular  to  6  and  OC  perpendicular  to  c,  these  intersecting 
at  C.  Complete  the  parallelogram  ABCQ"  and  join  P"Q"  and 
Q"0;  then  in  the  acceleration  diagram  OP"  =  a",  P"Q"  =  V 
and  Q"0  =  c"  which  gives  the  vector  acceleration  diagram  for 
the  part  OPQR.  Then  starting  at  Q",  which  gives  the  accelera- 

c/2 
tion  of  Q  on  the  vector  diagram,  draw  Q"D  =  —  and  parallel 

0 

T'  S/2 
to  e;  this  is  followed  by  DE  parallel  to  TS  and  of  length  — ~~-  = 

9 
f 

'  ,  and  the  vector  diagram  is  closed  by  drawing  EF  perpendicular 

to  c  to  meet  OF  perpendicular  to'/  in  F.  On  completing  the  par- 
allelogram DEFS",  the  point  S"  is  found  and  then  S"  is  joined 
to  0  and  to  Q".  The  line  S"Q"  represents  e  on  the  acceleration 
diagram  while  OS"  =  f  represents  /  on  the  same  figure. 

The  length  OS"  represents  the  acceleration  of  S  on  a  scale  of 
—  co2 : 1  and  the  acceleration  of  any  other  point  on  /  is  found  by 
locating  on  OS"  or  R"S"  a  point  similarly  situated  to  the  desired 
point  on  ST.  If  the  angular  acceleration  of  the  jaw  is  requiredv 
it  may  be  found  as  described  at  Sec.  229  and  evidently  is  a/  = 

~FOXT 

Calling  G  the  center  of  gravity  of  the  jaw  /  and  locating  G" 
in  the  same  way  with  regard  to  S"T"  that  G  is  located  with  regard 

19 


290 


THE  THEORY  OF  MACHINES 


to  ST,  the  acceleration  of  G  is  G"0  X  co2  and  the  force  required 
to  cause  this  acceleration  and  therefore  shaking  the  machine  is 


parallel  to  G"0  and  is  equal  to  G"0  X  co2  X 


weight  of  jaw 
32.16 


pds. 


Or  the  torque  required  for  the  purpose  is  //  X  a/  where  //  is  the 
moment  of  inertia  of  /  with  regard  to  G. 

233.  Application  to  the  Engine. — This  construction  and  the 
determination  of  the  accelerations  and  forces  has  a  very  useful 
application  in  the  case  of  the  reciprocating  engine  and  this  ma- 
chine will  now  be  taken  up.  Fig.  169  represents  an  engine  in 


FIG.  169. 

which  0  is  the  crankshaft,  P  the  crankpin  and  Q  the  wristpin,  the 
block  c  representing  the  crosshead,  piston  and  piston  rod.  Let 
the  crank  turn  with  angular  velocity  co  radians  per  second  and 
have  an  acceleration  a  in  the  sense  shown,  and  let  G  be  the  center 
of  gravity  of  the  connecting  rod  b.  To  get  the  vector  accelera- 
tion diagram  find  P"  exactly  as  in  the  former  construction,  OP 
representing  the  acceleration  PO.co2  and  PP"  the  acceleration  aa, 
both  on  the  scale  -co2  to  1. 

Now  the  motion  of  Q  is  one  of  sliding  and  thus  Q  has  only 
tangential  acceleration,  or  acceleration  in  the  direction  of  sliding, 
in  this  case  QS,  the  sense  being  determined  later.  Hence,  the 
total  acceleration  of  Q  must  be  represented  by  a  line  through  O 
in  the  direction,  QS  therefore  Q"  lies  on  a  line  through  the  center 
of  the  crankshaft,  and  the  diagram  is  reduced  to  a  simpler  form 


ACCELERATIONS  IN  MACHINERY  291 

than  in  the  more  general  case.     Having  found  P",  draw  P" 'A 

7/2 

parallel  to  6,  of  length  -j->  to  represent  QN,  and  also  draw  AQ", 

normal  to  P"A,  to  meet  the  line  Q"0,  which  is  parallel  to  QS, 
in  Q".  Then  will  AQ"  represent  the  value  of  the  angular  accel- 

n 

eration  of  the  rod  6.     Since  bab  =  Q"A.co2  or  a&  =  Q"A.j-}  and 

since  AQ"  lies  on  the  same  side  of  P"A  that  PP"  does  of  OP, 
therefore  a&  is  in  the  same  sense  as  a]  thus  since  co&  is  opposite  to 
o>,  the  angular  velocity  of  the  rod  is  decreasing,  or  the  rod  is  being 
retarded. 

The  acceleration  of  the  center  of  gravity  of  b  is  represented  by 
OG"  and  is  equal  to(r"0.co2,  and  similarly  the  acceleration  of  the 
end  Q  of  the  rod  is  represented  by  OQ"  and  is  equal  to  <2"0.o>2, 
this  being  also  the  acceleration  of  the  piston. 

It  will  be  observed  that  all  of  these  accelerations  increase  as 
the  square  of  the  number  of  revolutions  per  minute  of  the  crank- 
shaft, so  that  while  in  slow-speed  engines  the  inertia  forces  may 
not  produce  any  very  serious  troubles,  yet  in  high-speed  engines 
they  are  very  important  and  in  the  case  of  such  engines  as  are 
used  on  automobiles,  which  run  at  as  high  speeds  as  1,500  revo- 
lutions per  minute,  these  accelerations  are  very  large  and  the 
forces  necessary  to  produce  them  cause  considerable  disturb- 
ances. Take  the  piston  for  example,  the  force  required  to  move 
it  will  depend  on  the  product  of  its  weight  and  its  acceleration, 
so  that  if  an  engine  ran  normally  at  750  revolutions  per  minute 
and  then  it  was  afterward  decided  to  speed  it  up  to  1,500  revo- 
lutions per  minute,  the  force  required  to  move  the  piston  in  any 
position  in  the  latter  case  would  be  four  times  as  great  as  in  the 
former  case. 

234.  Approximate  Construction. — In  the  actual  case  of  the 
engine,  the  calculations  may  be  very  much  simplified  owing  to 
certain  limitations  which  are  imposed  on  all  designs  of  engines 
driving  other  machinery,  these  limitations  being  briefly  that  the 
variations  in  velocity  of  the  flywheel  must  be  comparatively 
small,  that  is,  the  angular  acceleration  of  the  flywheel  must  not 
be  great,  and  in  fact,  on  engines  the  flywheels  are  made  so  heavy 
that  a.  cannot  be  large. 

To  get  a  definite  idea  on  this  subject  a  case  was  worked  out  for 
a  10  by  10-in.  steam  engine,  running  at  310  revolutions  per  min- 
ute, and  the  maximum  angular  acceleration  of  the  crank  was 


292 


THE  THEORY  OF  MACHINES 


found  to  be  slightly  less  than  7  radians  per  second  per  second. 
For  this  case  the  normal  acceleration  of  P  is  rco2  =  ^{2  X  1,100  = 
458  ft.  per  second  per  second,  while  the  tangential  acceleration 

is  ra  =  y~  X  7  =  5.8  ft.  per  second  per  second,  which  is  very 

small  compared  with  458  ft.  per  second  per  second,  so  that  on 
any  ordinary  drawing  the  point  P"  would  be  very  close  to  P. 
Thus  without  serious  error  ra.  may  be  neglected  compared  with 
rcc2  and  hence  P"  is  at  P. 

With  the  foregoing  modification  for  the  engine,  the  complete 
acceleration  diagram  is  shown  at  Fig.  170,  the  length  PA  repre- 


W"' 


FIG.  170.  —  Piston  acceleration. 


sen  ting  -j-  and  AQ"  is  normal  to  b,  thus  P"Q"  is  the  acceleration 

diagram  for  the  connecting  rod  and  OQ  "  represents  the  accelera- 
tion of  the  piston  on  the  scale  -co2  to  1.  Two  cases  are  shown: 
(a)  for  the  ordinary  construction;  and  (&)  for  the  offset  cylinder. 
The  acceleration  of  any  such  point  as  G  is  found  by  finding  G", 
making  the  line  GG"  parallel  to  QQ",  the  acceleration  then  is 
G"0.co2. 

It  should  be  noticed  that  the  greater  the  ratio  of  6  to  a,  that 
is  the  longer  the  connecting  rod  for  a  given  crank  radius,  the 
more  nearly  will  the  point  A  approach  to  P  because  the  distance 

o 

PA  represents  the  ratio  -v-  and  this  steadily  decreases  as  b  in- 

creases, and  at  the  same  time  AQ"  becomes  more  nearly  vertical. 
In  the  extreme  case  of  an  infinitely  long  rod,  carried  out  practi- 
cally as  shown  at  Fig.  6,  the  point  A  coincides  with  P  and 


ACCELERATIONS  IN  MACHINERY  293 

is  vertical  and  then  the  acceleration  of  the  piston  which  is  OQ" 
is  simply  the  projection  of  a  on  the  line  of  the  piston  travel  or 
the  acceleration  Q"0  X  co2  =  a  .  cos  6  X  co2  where  0  is  the  crank 
angle  POQ". 

235.  Piston  Acceleration  at  Certain  Points.  —  Taking  the  more 
common  form  of  the  mechanism  shown  at  Fig.  170  (a)  the  num- 
erical values  of  the  acceleration  of  the  piston  may  be  found  in  a 
few  special  cases.  When  the  crank  is  vertical,  bf  is  zero  and  there- 
fore A  is  at  P  vertically  above  0,  so  that  when  AQ"  is  drawn, 
Q"  lies  to  the  left  of  0  showing  that  the  piston  has  negative  ac- 
celeration or  is  being  retarded.  For  this  position  a  circle  of  diam- 
eter QQ"  will  pass  through  P  and  therefore  Q"0  X  OQ  =  OP2  or 

OP2  a2 

Q"0  =  ~7^  =  —  jr^          and  the  acceleration  of  the  piston  is 


a 


o>2  X      /ry~^^2  ^'  Per  second  Per  second. 

fr'2       a2 

At  both  the  dead  centers  bf  =  a  hence  P"A  =  -7-  =  ~r>  so 

6         6 

a2 

that  for  the  head  end,  Q"0  =  a  +  -r-  and  the  piston  has  its  maxi- 
mum acceleration  at  this  point,  which  is  (a  +  ~r ]  <o2  toward  O, 
while  for  the  crank  end,  Q"O  =  a  --  -r  and  the  acceleration  is 

(a j-\  co2  toward  0,  or  the  piston  is  being  retarded. 

Example. — Let  an  engine  with  7-in.  stroke  and  a  connecting 
rod  18  in.  long  run  at  525  revolutions  per  minute.     Then  a  = 

3/^  18 

-TTT  =  0.29  ft.,  b  =  -r~o  =  1.5  ft.,  and  co  =  55  radians  per  second, 
-i-^j  \.2i 

At  the  head  end  the  acceleration  of  the  piston  would  be: 

(o>2\  I  0  29  2\ 

a+  -T-)  co2  =  (0.29  +  -4-=-)  X  552  =  931    ft.    per   second   per 
0 /  \  l.O  / 

second. 

At  the  crank  end  the  acceleration  would  be: 

(a  -  ~)  co2  =  (o.29  -    -J-R-)  X  552  =  623    ft.   per    second  per 

second. 

At  the  time  when  the  crank  is  vertical  the  result  is: 

X  552  =  173  ft.  per  second  per 


294 


THE  THEORY  OF  MACHINES 


The  angular  acceleration  of  the  connecting  rod,  being  deter- 
mined by  the  length  AQ",  is  zero  at  each  of  the  dead  points  but 
when  the  crank  is  vertical  it  has  nearly  its  maximum  value;  the 

formula  for  it  is  Q"A  X  y.     When  the  crank  is  vertical  a  dia- 
gram will  show  that 

n^ 

Q"A 

and  the  acceleration  is 


Vb2-  a2 


For  the  engine  already  examined,  when  in  this  position, 

[0  292       1 
=  552  =  596  radians  per  second  per  second. 
Vl.52  -  0.292J 

APPROXIMATE  GRAPHICAL  SOLUTION  FOR  THE  STEAM  ENGINE 
236.  In  the  approximate  method  already  described,  in  which 


FIG.  171. 


the  angular  acceleration  of  the  crankshaft  is  neglected  and  P" 
is  assumed  to  coincide  with  P,  it  will  be  noticed  that  the  length 

o 

P"A  =  -T-,  is  laid  off  along  the  connecting  rod,  the  length  P'Q' 


ACCELERATIONS  IN  MACHINERY  295 

representing  &',  and  PQ  the  length  b,  and  then  AQ"  is  drawn  per- 
pendicular to  PQ.  This  may  be  carried  out  by  a  very  simple 
graphical  method  as  follows:  With  center  P  and  radius  P'Q' 
=  b'  describe  a  circle,  Fig.  171,  also  describe  a  second  circle, 
having  the  connecting  rod  b  as  its  diameter,  cutting  the  first 
circle  at  M  and  N,  and  join  MN.  Where  MN  cuts  b  locates 
the  point  A  and  where  it  cuts  the  line  through  0  in  the  direction 
of  motion  of  Q  gives  Q". 

The  proof  is  that  PMQ  being  the  angle  in  a  semicircle  is  a  right 
angle  also  the  chord  MN  is  normal  to  PQ  and  is  bisected  at  A. 
Then  in  the  circle  MPNQ  there  are  two  chords  PQ  and  MN  inter- 
secting at  A,  and  hence  from  geometry  it  is  known  that: 

PA.AQ  =  MA.AN  =  MA2 

or 

PA(PQ  -  PA)  =  MP2  -  PA2  =  b'    -  PA2. 

Multiplying  out  the  left-hand  side  and  cancelling 

PA.PQ  =  b'2 
that  is 


PQ~    b 

which  proves  the  construction. 

THE  EFFECTS  OF  THE  ACCELERATIONS  OF  THE  PARTS  UPON  THE 
FORCES  ACTING  AT  THE  CRANKSHAFT  OF  AN  ENGINE 

In  order  to  accelerate  or  retard  the  various  parts  of  the  engine, 
some  torque  must  be  required  or  will  be  produced  at  the  crank- 
shaft, and  a  study  of  this  will  now  be  taken  up  in  detail. 

237.  (a)  The  effect  produced  by  the  piston. 

By  the  construction  already  described  the  acceleration  of  the 
piston  is  readily  found  and  it  will  be  seen  that  Q"  lies  first  on  the 
cylinder  side  of  0  and  then  on  the  opposite  side.  When  Q"  lies 
between  0  and  Q,  Fig.  172,  then  the  acceleration  of  the  piston  is 
Q"O  X  co2,  and  the  acceleration  of  the  piston  is  in  the  same  sense 
as  the  motion  of  the  piston,  or  the  piston  is  being  accelerated. 
Conversely,  when  Q"  lies  on  QO  produced  the  acceleration  being 
in  the  opposite  sense  to  the  motion  of  the  piston,  the  latter  is 
being  retarded.  These  statements  apply  to  the  motion  of  the 
piston  from  right  to  left,  when  the  sense  of  motion  of  the 
piston  reverses  the  remarks  about  the  acceleration  must  also 
be  changed.  If  now  the  accelerations  for  the  different 


296 


THE  THEORY  OF  MACHINES 


piston  positions  on  the  forward  stroke  be  plotted,  the  diagram 
EJH  will  be  obtained,  Fig.  172,  where  the  part  of  the  diagram 
EJ  represents  positive  accelerations  of  the  piston,  and  the  part 
JH  negative  accelerations,  or  retardations.  The  corresponding 
diagram  for  the  return  stroke  of  the  piston  is  omitted  to  avoid 
complexity. 


E 


FIG.  172. — Acceleration  of  piston  on  forward  stroke. 

Let  the  combined  weight  of  the  piston,  piston  rod  and  cross- 
head  be  wc  pounds,  the  corresponding  mass  being  mc  =  — ,  and 

\y 

let  /  represent  the  acceleration  of  the  piston  at  any  instant ;  then 
the  force  Pc  necessary  to  produce  this  acceleration  is  Pc  =  rac ./. 


FIG.  173. — Modification  of  diagram  due  to  inertia  of  piston. 

This  force  will  be  positive  if  /  is  positive  and  vice  versa,  that  is,  if  / 
is  positive  a  force  must  be  exerted  on  the  piston  in  its  direction  of 
motion  and  if  it  is  negative  the  force  must  be  opposed  to  the 
motion.  In  the  first  case  energy  must  be  supplied  by  the  flywheel, 
or  steam,  or  gas  pressure,  to  speed  up  the  piston,  whereas,  in  the 
latter  case,  energy  will  be  given  up  to  the  flywheel  due  to  the 


ACCELERATIONS  IN  MACHINERY  297 

decreasing  velocity  of  the  piston.  Since  no  net  energy  is  received 
during  the  operation,  therefore,  the  work  done  on  the  piston 
in  accelerating  it  must  be  equal  to  that  done  by  the  piston  while 
it  is  being  retarded. 

Two  methods  are  employed  for  finding  the  turning  effect  of  this 

force,  Pc]  the  first  is  to  reduce  it  to  an  equivalent  amount  per 

p 

square  inch  of  piston  area  by  the  formula  pc  =  -r  where  A  is 

A. 

the  area  of  the  piston,  and  then  to  correct  the  corresponding  pres- 
sures shown  by  the  indicator  diagram  by  this  amount.  In  this 
way  a  reduced  indicator  diagram  for  each  end  is  found,  as  shown 
for  a  steam  engine  in  Fig.  173,  where  the  dotted  diagram  is  the 
reduced  diagram  found  by  subtracting  the  quantity  pc  from  the 
upper  line  on  each  diagram.  The  remaining  area  is  the  part 
effective  in  producing  a  turning  moment  on  the  crankshaft. 

The  second  method  is  to  find  directly  the  turning  effect  neces- 
sary on  the  crankshaft  to  overcome  the  force  Pc,  and  from  the 
principles  of  the  phorograph  this  torque  is  evidently  Tc  = 
PC  X  OQ'  =  mc  X  /  X  OQ'.  In  the  position  shown  in  Fig. 
172,  Pe  would  act  as  shown,  and  a  torque  acting  in  the  same 
sense  as  the  motion  of  a  would  have  to  be  applied. 

The  first  method  is  very  instructive  in  that  it  shows  that  the 
force  necessary  to  accelerate  the  piston  at  the  beginning  of  the 
stroke  in  very  high-speed  engines  may  be  greater  than  that  pro- 
duced by  the  steam  or  gas  pressure,  and  hence,  that  in  such  cases 
the  connecting  rod  may  be  in  tension  at  the  beginning  of  the 
stroke,  but,  of  course,  before  the  stroke  has  very  much  proceeded 
it  is  in  compression  again.  This  change  in  the  condition  of  stress 
in  the  rod  frequently  causes  " pounding"  due  to  the  slight  slack- 
ness allowed  at  the  various  pins. 

238.  (6)  The  Effect  Produced  by  the  Connecting  Rod.— 
This  effect  is  rather  more  difficult  to  deal  with  on  account  of 
the  nature  of  the  motion  of  the  rod.  The  resultant  force  acting 
may,  however,  be  found  by  the  method  described  earlier  in  the 
chapter,  Sec.  230,  but  in  the  case  of  the  engine,  the  construction 
may  be  much  simplified,  and  on  account  of  the  importance  of  the 
problem  the  simpler  method  will  be  described  here.  It  consists 
in  dividing  the  rod  up  into  two  equivalent  concentrated  masses, 
one  at  the  crosshead  pin  the  other  at  a  point  to  be  determined. 

Referring  to  Fig.  174,  the  rod  is  represented  on  the  acceleration 
diagram  by  P"Q"  and  the  acceleration  of  any  point  on  it  or  the 


298 


THE  THEORY  OF  MACHINES 


angular  acceleration  of  the  rod  may  be  found  by  processes 
already  explained.  Let  h  be  the  moment  of  inertia  of  the  rod 
about  its  center  of  gravity,  kb  being  the  corresponding  radius  of 
gyration  and  nib  the  mass,  so  that  7&  =  ra&fcb2,  and  let  the  center 
of  gravity  G  lie  on  PQ  at  distance  TI  from  Q.  Instead  of  consider- 
ing the  actual  rod  it  is  possible  to  substitute  for  it  two  concen- 
trated masses  mi  and  m2,  which,  if  properly  placed,  and  if  of 
proper  weight,  will  have  the  same  inertia  and  weight  as  the 

original  rod.     Let  these  masses  be  nil  and  m2  where  m\.  =  — 

and  m2  =  — 'in  which  w\  and  w2  are  the  weights  of  the  masses  in 
pounds.  Further,  let  mass  m\  be  concentrated  at  Q,  it  is  required 


FIG.  174. 

to  find  the  weights  Wi  and  w2  and  the  position  of  the  weight  w2. 
Let  r2  be  the  distance  from  the  center  of  gravity  of  the  rod  to 
mass  m2. 

These  masses  are  determined  by  the  following  three  conditions : 

1.  The  sum  of  the  weights  of  the  two  masses  must  be  equal 
to  the  weight  of  the  rod,  that  is,  Wi  +  w2  =  Wb,  or  nil  +  m2  =  nib. 

2.  The  two  masses  m\  and  m2,  must  have  their  combined 
center  of  gravity  in  the  same  place  as  before;  therefore,  m\r\  = 
m2r2. 

3.  The  two  masses  must  have  the  same  moment  of  inertia 
about  their  combined  center  of  gravity  G  as  the  original  rod  has 
about  the  same  point;  hence 

-\-  m2r22    = 


CD 

(2) 
(3) 


For  convenience  these   are  assembled   here: 
nil  -\-  m2  =  mb 


=  m2r2 


-f- 


ACCELERATIONS  IN  MACHINERY  299 

Solving  these  gives: 

mi  =  mb  X  ~  and    w2  =  mb  X  - 

7*1  -r  7*2  ri  ~r  r2 

7    2  ^2 

and  rir2  =  fc&2   or  r2  =  — 

Thus,  for  the  purposes  of  this  problem  the  whole  rod  may  be 
replaced  by  the  two  masses  mi  and  w2  placed  as  shown  in  Fig.  174. 
The  one  mass  mi  merely  has  the  same  effect  as  an  increase  in  the 
weight  of  the  piston  and  the  method  of  finding  the  force  required 
to  accelerate  it  has  already  been  described.  Turning  then  to  the 
mass  ra2,  which  is  at  a  fixed  distance  r2  from  G]  the  center  of  grav- 
ity of  m2  is  K  and  the  acceleration  of  K  is  evidently  K"0  X  co2, 
K"K  being  parallel  to  G"G.  The  direction  of  the  force  acting 
on  w2is  the  same  as  that  of  the  acceleration  of  its  center  of  gravity 
and  is  therefore  parallel  to  K/fO,  and  the  magnitude  of  this  force 
is  w2  X  K"0  X  co2.  The  force  acts  through  K,  its  line  of  action 
being  KL  parallel  to  K"0. 

The  whole  rod  may  now  be  replaced  by  the  two  masses  m\ 
and  w2.  The  force. acting  on  the  former  is  m\  X  Q"0  X  co2 
through  Q  parallel  to  Q"0,  that  is,  this  force  is  in  the  direction  of 
motion  of  Q  and  passes  through  L  on  Q"0.  The  force  on  the 
mass  ra2  is  ra2  X  KfrO  X  co2,  which  also  passes  through  L,  so 
that  the  resultant  force  F  acting  on  the  rod  must  also  pass  through 
L.  Thus  the  construction  just  described  gives  a  convenient 
graphical  method  for  locating  one  point  L  on  the  line  of  action 
of  the  resultant  force  F  acting  on  the  connecting  rod. 

Having  found  the  point  L  the  direction  of  the  force  F  has  been 
already  shown  to  be  parallel  to  G"0  and  its  magnitude  is 
mb  X  G"0  X  co2.  Let  F  intersect  the  axis  of  the  rod  at  H,  find 
the  image  Hf  of  H,  and  transfer  F  to  H1 '.  The  moment  required 
to  produce  the  acceleration  of  the  rod  is  then  Fh. 

A  number  of  trials  on  different  forms  and  proportions  of  en- 
gines have  shown  that  the  point  L  remains  in  the  same  position 
for  all  crank  angles,  and  hence  if  this  is  determined  once  for  a 
given  engine  it  will  be  only  necessary  to  determine  G"0  for  the 
different  crank  positions;  as  this  enables  the  magnitude  and 
direction  of  F  to  be  found  and  its  position  is  fixed  by  the 
point  L. 

239.  Net  Turning  Moment  on  Crankshaft. — For  the  position 
of  the  machine  shown  in  Fig.  174,  let  P  be  the  total  pressure  on 


300  THE  THEORY  OF  MACHINES 

the  piston  due  to  the  gas  or  steam  pressure;  then  the  net  turning 
moment  acting  on  the  crankshaft  is 

P  X  OQ'  -  [mc  X  Q"0  X  co2  X  OQf  +  mb  X  G"0  X  w2  X  h] 

after  allowance  has  been  made  for  the  inertia  of  the  piston 
and  connecting  rod.  This  turning  moment  will  produce  an  ac- 
celeration or  retardation  of  the  flywheel  according  as  it  exceeds 
or  is  less  than  the  torque  necessary  to  deliver  the  output. 

All  of  these  quantities  have  been  determined  for  the  complete 
revolution  of  a  steam  engine  and  the  results  are  given  and  dis- 
cussed at  the  end  of  the  present  chapter. 

240.  The  Forces  Acting  at  the  Bearings. — The  methods  de- 
scribed enable  the  pressures  acting  on  the  bearings  due  to  the 
inertia  forces  to  be  easily  determined,  and  this  problem  is  left  for 
the  reader  to  solve  for  himself. 

In  high-speed  machinery  the  pressures  on  the  bearings  due 
to  the  inertia  of  the  parts  may  become  very  great  indeed  and  all 
care  is  taken  by  designers  to  decrease  them.  Thus,  in  automobile 
engines,  some  of  which  attain  as  high  a  speed  as  3,000  revolutions 
per  minute,  or  over,  during  test  conditions,  the  rods  are  made  as 
light  as  possible  and  the  pistons  are  made  of  aluminum  alloy  in 
order  to  decrease  their  weight.  In  one  of  the  recent  automobile 
engines  of  3-in.  bore  and  5-in.  stroke  the  piston  weighs  17  oz. 
and  the  force  necessary  to  accelerate  the  piston  at  the  end  of  the 
stroke  and  at  a  speed  of  3,000  revolutions  per  minute  is  over  800 
pds.,  corresponding  to  an  average  pressure  of  over  110  pds.  per 
square  inch  on  the  piston  and  the  effect  of  the  connecting  rod 
would  increase  this  approximately  50  per  cent;  thus  during  the 
suction  stroke  the  tension  in  the  rod  is  over  1,200  pds.  at  the  head- 
end dead  center  and  the  compressive  stress  in  the  rod  is  much 
less  than  that  corresponding  to  the  gas  pressure.  At  the  crank- 
end  dead  center  the  accelerating  force  is  also  high,  though  less 
than  at  the  head  end,  and  here  also  the  rod  is  in  compression  due 
to  the  inertia  forces.  If  the  gas  pressure  alone  were  considered, 
the  rod  would  be  in  compression  in  all  but  the  suction  stroke. 

241.  Computation    on  an  Actual   Steam  Engine. — In  order 
that  the  methods  may  be  clearly  understood  an  example  is  worked 
out  here  of  an  engine  running  at  525  revolutions  per  minute, 
and  of  the  vertical,  cross-compound  type  with  cranks  at  180°, 
and  developing  125  hp.  at  full  load.     Both  cylinders  are  7  in. 
stroke  and  11  in.  and   15 J^  in.  diameter  for  the  high-  and  low- 


ACCELERATIONS  IN  MACHINERY 


301 


pressure  sides  respectively.  The  weight  of  each  set  of  recip- 
rocating parts  including  piston,  piston  rod  and  crosshead  is 
161  lb.,  while  the  connecting  rod  weighs  47  Ib.  has  a  length 
between  centers  of  18  in.  and  its  radius  of  gyration  about  its  cen- 


FIG.  175. 


ter  of  gravity  is  7.56  in.,  the  latter  point  being  located  13.3  in. 
from  the  center  of  the  wristpin. 

From  the  above  data  o>  =  55  radians  per  second, 


161 
32.2 
13.3 


5,   mb  = 


=  1.46    and    7*  =  -        =  0.63    ft. 


Alsor1=  — ^-  =  1.11  ft.,  r2  = 


12 


1.46  X 


0.632 
1.11 
0.36 


1.11  +  0.36 


0.36  ft.  and 


-  =  0.35  while  mz  =  1.11. 


800 
600 
„  400 

•d 
§200 

I  ° 

1/1 

"\ 

^ 

tal  Torque  E< 

quired 

^ 



/ 

-Torque  Requ 
Beciprocatin 

red  Ifor 

Patts 

/ 

\ 

^ 

. 

^—  —  — 

\ 
\    90     1 

)8    1 

6    144    1 

32 

/ 



\2 

Crank  Angles 
38    306    324   342  36 

J 

8     3 

6      5 

4      1 

m 

Connecting 

BRdol° 

f/ 

80  1 

J8    2 

(>    2, 

i4    2, 

U    2" 

5 

—    "•* 

V 

g  ^ 

600 

800 

- 

\ 

/ 

f 

\\ 

/ 

^r 

A 

'/ 





— 

A 

/ 

V 

>/ 

FIG.  176. — Effect  of  connecting  rod  and  piston. 

The  construction  for  the  crank  angle  36°  is  shown  in  Fig.  175 
with  all  dimensions  marked  on,  and  the  complete  results  for  the 
entire  revolution  for  one  side  of  the  engine  are  set  down  on  the 
accompanying  table,  all  of  the  quantities  being  tabulated.  The 
point  L  for  this  engine  is  located  0.44  in.  from  0  and  on  the 
cylinder  side  of  it.  The  table  shows  that  at  the  head  end  a 
force  of  5,262  lb.  would  be  required  to  accelerate  each  piston 
which  corresponds  to  a  mean  pressure  for  the  high-pressure 


302 


THE  THEORY  OF  MACHINES 


side  of  55  Ib.  per  square  inch.,  in  other  words  if  the  net  steam 
pressure  fell  below  55  Ib.  at  this  point  the  high-pressure  rod 
would  be  in  tension  instead  of  compression. 

The  disturbing  effect  of  the  connecting  rod  is  much  less  marked 
as  the  table  shows,  but  in  accurate  calculations  cannot  be  neg- 
lected. The  combined  effect  of  the  two  as  shown  in  the  last 
column  is  quite  decided. 

In  order  that  the  results  may  be  more  clearly  understood 
they  have  been  plotted  in  Fig.  176,  which  shows  the  turning 
moment  at  the  crankshaft  required  to  move  the  piston  and  the 
crosshead  separately,  and  also  the  combined  effort  required  for 
both.  The  turning  effort  required  for  the  rod  is  not  quite  one- 
twelfth  that  required  for  the  piston. 


TABLE  SHOWING  THE  EFFECT  DUE  TO  THE  INERTIA  OF  THE  PARTS  OF  AN 

11  BY  7-iN.  STEAM  ENGINE  RUNNING  AT  525  REVOLUTIONS 

PER  MINUTE 


tt 

Piston,  crosshead,  etc. 

Connecting  rod 

Total  turn- 
ing moment 

* 

reouired 

•s 

i 

. 

G-rt 

05 

at  crank  to 

a 

03 

'3,  § 

o 

a 

'O 

move   all 

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0 

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parts, 

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+  711 

0.295 

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0.217 

656 

958 

0.023 

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232 

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-0.060 

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-908 

0.292 

-265 

0.215 

650 

950 

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-22 

287 

108 

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414 

2,072 

0.260 

539 

0.228 

690 

1,007 

0.045 

45 

584 

126 

0.187 

566 

2,829 

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588 

0.248 

750 

1,095 

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656 

3,282 

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469 

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1,179 

0.035 

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0.273 

824 

1,205 

0.017 

+  20 

276 

216 

0.217 

656 

3,282 

0.143 

469 

0.267 

808 

1,179 

0.035 

41 

510 

234 

0.187 

566 

2,829 

0.208 

588 

0.248 

750 

1,095 

0.048 

48 

636 

252 

0.137 

414 

2,072 

0.260 

539 

0.228 

690 

1,007 

0.045 

45 

584 

270 

+0.060 

+  181 

+908 

0  292 

+  265 

0.215 

650 

950 

0.023 

+22 

287 

288 

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-711 

0.295 

-210 

0.217 

656 

958 

0.023 

-22 

232 

306 

0.153 

463 

2,314 

0.264 

611 

0.242 

732 

1,069 

0.050 

53 

664 

324 

0.253 

765 

3,827 

0.198 

758 

0.272 

823 

1,201 

0.052 

62 

820 

342 

0.325 

983 

4,916 

0.107 

-526 

0.297 

898 

1,311 

0.033 

-43 

569 

360 

-0.348 

-1,053 

-5,262 

0 

0 

0.349 

1,056 

1,542 

0 

0 

0 

ACCELERATIONS  IN  MACHINERY 


303 


The  relative  effects  of  these  turning  moments  is  shown  more 
clearly  at  Fig.  177  in  which  separate  curves  are  drawn  for  the 
high-  and  low-pressure  sides.  The  dotted  curves  in  both  cases 
show  the  torque  due  to  the  steam  pressure  found  as  in  Chapter  X, 
while  the  broken  lines  show  the  torque  required  to  accelerate  the 
parts  and  the  curves  in  solid  lines  indicate  the  net  resultant 
torque  acting  on  the  crankshaft.  The  reader  will  be  at  once 


,-1200 

2 

£  80° 

§400 
H 

0 
400 
800 
1200 
1200 
*  800 

£ 

I  400 

2 

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FIG.  177. — Torque  diagrams  allowing  for  inertia  of  parts. 

struck  with  the  modification  produced  by  the  inertia  of  the  parts, 
but  it  must  always  be  kept  in  mind  that  this  only  modifies  the 
result  but  produces  no  net  change,  as  the  energy  used  up  in 
accelerating  the  masses  for  one  part  of  the  revolution  is  returned 
when  the  masses  are  retarded  later  on  in  the  cycle  of  the  ma- 
chine. That  these  forces  must  be  reckoned  with,  especially  in 
high-speed  machinery,  is  very  evident. 

242.  The  Gnome  Motor. — One  further  illustration  of  the 
principles  stated  here  may  be  given  in  the  Gnome  motor,  which 
has  had  much  application  in  aeroplane  work.  The  general  form 
of  the  motor  has  already  been  shown  in  Fig.  12  in  the  early 


304 


THE  THEORY  OF  MACHINES 


part  of  this  book  and  it  has  been  explained  that  the  mechanism 
is  exactly  the  same  as  in  the  ordinary  reciprocating  engine 
except  that  the  crank  is  fixed  and  the  connecting  rod,  cylinder  and 
other  parts  make  complete  revolutions.  The  mechanism  is  shown 
in  Fig.  178  in  which  a  is  the  cylinder  and  parts  secured  to  it,  6  is 
the  connecting  rod  and  c  is  the  piston,  and  power  is  delivered 
from- the  rotating  link  a,  which  is  assumed  to  turn  at  constant 
speed  of  co  radians  per  second. 

At  the  wristpin  two  letters  are  placed,  Q  on  the  rod  b  and  P 
on  a  directly  below  Q,  and  thus  as  the  revolution  proceeds  P 


FiG.  178. — Gnome  motor. 

moves  in  and  out  along  a;  the  motion  of  Q  relative  to  P  must,  in 
the  nature  of  the  case,  be  one  of  sliding  in  the  direction  of  a. 
The  phorograph  is  obtained  in  a  similar  way  to  that  for  the 
Whitworth  quick-return  motion,  Fig.  38,  the  only  difference  here 
being  that  the  cylinder  link  a  turns  at  uniform  speed  while  in 
the  former  case  the  connecting  rod  did  so.  The  image  Q'  lies 
on  P'Q'  normal  to  a  and  on  R'Q'  through  0  parallel  to  b. 

To  find  the  acceleration  diagram  the  plan  followed  in  Sec. 
228  is  employed.  Thus,  the  acceleration  of  R  relative  to  Q  added 
to  that  of  Q  relative  to  P  and  that  of  P  relative  to  0  must 
be  zero.  Using  the  notation  of  Sec.  228  it  follows  that  RT  + 
RN  +  QT  +  QN  +  PT  +  PN  =  0;  and  since  the  only  acceleration 


ACCELERATIONS  IN  MACHINERY  305 

which  Q  can  have  relative  to  P  is  tangential,  it  follows  that  QN  = 
0.  Again,  since  a  turns  at  a  uniform  speed,  the  value  of  PT  is 
also  zero.  Hence,  the  result  is  RT  +  RN  +  QT  +  PN  =  0. 

Now,  adopting  the  scale  of  —  co2: 1,  the  acceleration  PN  =  aco2 
is  represented  by  OP"  =  a  and  it  has  also  been  shown  in  Sec.  228 

fc'2 

that  RN  =  -j-  X  co2,  so  that  P"  B  is  laid  off  along  b  and  equal  to 

6'2 

-T-,  and  thus  P"#  will  represent  RN.     The  vector  diagram  is  closed 

by  RT  and  QT,  the  former  perpendicular  to  6  and  the  latter  parallel 
to  a,  so  that  BC  perpendicular  to  b  represents  RT  and  hence  CR" 
=  QT-  It  will  make  a  more  correct  vector  diagram  to  lay  off 
p"Q"  =  CR"  and  make  Q"A  and  AR"  equal  respectively  to  P"B 
and  BC.  Then  OP"  =  P*,  P"Q"  =  Qr,  Q"A  =  #„,  AR»  =  RT 
and  Q".R"  represents  the  rod  b  vectorially  on  the  acceleration 
diagram,  G"  corresponding  to  its  center  of  gravity  G.  The  accel- 
eration of  the  center  of  gravity  G  of  b  is  G"0  X  co2  and  the  an- 

o 

gular  acceleration  of  the  rod  is  R"A  X  -j-  as  given  in  Sec.  229. 

The  pull  on  6  due  to  the  centrifugal  effect  of  the  piston  is  Q"0 

weight  of  position  . 

X  co2  X  -        — ^n~n —      ~~  ln  the  direction  of  a. 
oZ.Z 

The  resultant  force  F  on  the  rod  6  may  be  found  as  in  Sec.  230 
and  is  in  the  direction  G"0,  that  is,  parallel  to  b.  Its  position 
is  shown  on  the  figure  and  the  pressure  between  the  piston  and 
cylinder  due  to  this  force  is  readily  found  knowing  the  value  and 
position  of  F. 

QUESTIONS  ON  CHAPTER  XV 

1.  A  weight  of  10  Ib.  is  attached  by  a  rod  15  in.  long  to  a  shaft  rotating  at 
100  revolutions  per  minute;  find  the  acceleration  of  the  weight  and  the  ten- 
sion in  the  rod. 

2.  If  the  shaft  in  question  1  increases  in  speed  to  120  revolutions  per 
minute  in  40  sec.,  find  the  tangential  acceleration  of  the  weight  and  also  its 
total  acceleration. 

3.  A  railroad  train  weighing  400  tons  is  brought  to  rest  from  50  miles  per 
hour  in  1  mile.     Find  the  average  rate  of  retardation  and  the  mean  resistance 
used. 

4.  At  each  end  of  the  stroke  the  velocity  of  a  piston  is  zero;  how  is  its 
acceleration  a  maximum? 

6.  Weigh  and  measure  the  parts  of  an  automobile  engine  and  compute 
the  maximum  acceleration  of  the  parts  and  the  piston  pressure  necessary  to 
produce  it. 
20 


306  THE  THEORY  OF  MACHINES 

6.  Find  the  bending  stresses  in  the  connecting  rod  of  the  same  engine,  due 
to  inertia,  when  the  crank  and  rod  are  at  right  angles. 

7.  Divide  the  rod  in  question  6  up  into  its  equivalent  masses,  locating  one 
at  the  wristpin. 

8.  Make  a  complete  determination  for  an  automobile  engine  of  the  result- 
ing torque  diagram  due  to  the  indicator  diagram  and  inertia  of  parts. 


CHAPTER  XVI 
BALANCING  OF  MACHINERY 

243.  General  Discussion  on  Balancing. — In  all  machines 
the  parts  have  relative  motion,  as  discussed  in  Chapter  I.  Some 
of  the  parts  move  at  a  uniform  rate  of  speed,  such  as  a  crankshaft 
or  belt-wheel  or  flywheel,  while  other  parts,  such  as  the  piston, 
or  shear  blade  or  connecting  rod,  have  variable  motion.  The 
motion  of  any  of  these  parts  may  cause  the  machine  to  vibrate 
and  to  unduly  shake  its  foundation  or  the  building  or  vehicle 
in  which  it  is  used.  It  is  also  true  that  the  annoyance  caused  by 
this  vibration  may  be  out  of  all  proportion  to  the  vibration 
itself,  the  results  being  so  marked  in  some  cases  as  to  disturb 
buildings  many  blocks  away  from  the  place  where  the  machine 
is.  This  disturbance  is  frequently  of  a  very  serious  nature, 
sometimes  forcing  the  abandonment  of  the  faulty  machine  alto- 
gether; therefore  the  cause  of  vibration  in  machinery  is  worthy  of 
careful  examination. 

It  is  not  possible  in  the  present  treatise  to  discuss  the  general 
question  of  vibrations,  as  the  matter  is  too  extensive,  but  it 
may  be  stated  that  one  of  the  most  common  causes  is  lack  of 
balance  in  different  parts  of  the  machine  and  the  present  chapter 
is  devoted  entirely  to  the  problem  of  balancing.  Where  any 
of  the  links  in  a  machine  undergo  acceleration  forces  are  set  up 
in  the  machine  tending  to  shake  it,  and  unless  these  forces  are 
balanced,  vibrations  of  a  more  or  less  serious  nature  will  occur, 
but  balancing  need  only  be  applied  where  accelerations  of  the 
parts  occur. 

It  must  be  borne  in  mind,  however,  that  the  accelerations  are 
not  confined  solely  to  such  parts  as  the  piston  or  the  connecting 
rod  which  have  a  variable  motion,  but  the  particles  compos- 
ing any  mass  which  is  rotating  with  uniform  velocity  about 
a  fixed  center  also  have  acceleration1  and  may  throw  the 
machine  out  of  balance,  because,  as  explained  in  Sec.  226, 

1  In  connection  with  this  the  first  part  of  Chapter  XV  should  be  read 
over  again. 

307 


308  THE  THEORY  OF  MACHINES 

a  mass  has  acceleration  along  its  path  when  its  velocity  is 
changing,  and  also  acceleration  normal  to  its  curved  path 
even  when  its  velocity  is  constant.  In  discussing  the  sub- 
ject it  is  most  convenient  to  divide  the  problem  up  into  two 
parts,  dealing  first  with  links  which  rotate  about  a  fixed  center 
and  second  with  those  which  have  a  different  motion,  in  all 
cases  plane  motion  being  assumed. 

THE  BALANCING  OF  ROTATING  MASSES 
244.  Balancing  a  Single  Mass.  —  Let  a  weight  of  w  Ib.  which  has 

IV 

a  mass  m  =  ~  rotate  about  a  shaft  with  a  fixed  center,  at  a  fixed 

y 

radius  r  ft.,  and  let  the  radius  have  a  uniform  angular  velocity 
of  co  radians  per  second.  Then,  referring  to  Sec.  226  this  mass 
will  have  no  acceleration  along  its  path  since  co  is  assumed  con- 
stant, but  it  will  have  an  acceleration  toward  the  axis  of  rotation 
of  rco2  ft.  per  second  per  second,  and  hence  a  radial  force  of 

IV 

amount  —rco2  =  mrco2  pds.  must  be 

Q 

applied  to  it  to  maintain  it  at  the 
given  radius  r.  This  force  must  be 
applied  by  the  shaft  to  which  the 
weight  is  attached,  and  as  the 
weight  revolves  there  will  be  a  pull 
on  the  shaft,  always  in  the  radial 
direction  of  the  weight,  and  this 
pull  will  thus  produce  an  unbalanced 
FIG.  179.  force  on  the  shaft,  which  must  be 

balanced  if  vibration  is  to  be  avoided. 

Let  Fig.  179  represent  the  weight  under  consideration  in  one 
of  its  positions;  then  if  vibration  is  to  be  prevented  another 
weight  Wi  must  be  attached  to  the  same  shaft  so  that  its  accelera- 
tion will  be  always  in  the  same  direction  but  in  opposite  sense  to 
that  of  w,  and  this  is  possible  only  if  w\  is  placed  at  some  radius 
7*1  and  diametrically  opposite  to  w.  Clearly,  the  relation  be- 

tween the  two  weights  and  radii  is  given  by  —  rco2  =  —  rico2  or 
rw  =  riWi,  since  —  is  common  to  both  sides,  from  which  the 


product  riWi  is  found,  and  having  arbitrarily  selected  one  of 
these  quantities  such  as  rif  the  value  of  Wi  is  easily  determined. 


BALANCING  OF  MACHINERY 


309 


If  the  two  weights  are  placed  as  explained  there  will  be  no  resul- 
tant pull  on  the  shaft  during  rotation,  and  hence  no  vibration; 
in  other  words  the  shaft  with  its  weights  is  balanced. 

It  sometimes  happens  that  the  construction  prevents  the 
placing  of  the  balancing  mass  directly  opposite  to  the  weight  w, 
as  for  example  in  the  case  of  the  crankpin  of  an  engine,  and  then 
the  balancing  weights  must  be  divided  between  two  planes 
which  are  usually  on  opposite  sides  of  the  disturbing  mass, 
although  they  may  be  on  the  same  side  of  it  if  desired.  Let  Fig. 
180  represent  the  crankshaft  of  an  engine,  and  let  the  crankpin 
correspond  to  an  unbalanced  weight  w  Ib.  at  radius  r.  The  planes 
A  and  B  are  those  in  which  it  is  possible  to  place  counterbalance 
weights  and  the  magnitude  and  position  of  the  weights  are 


FIG.  180.  —  Crank-shaft  balancing. 

required.     Let  the  weights  be  w\  and  wz  Ib.  and  their  radii  of 
rotation  be  r\  and  r2  respectively;  then  clearly  the  vector  sum 

Let  all   the 


'—  wr)    -  - 


0,   or 


+  wzr2  =  wr. 


masses  be  in  the  plane  containing  the  axis  of  the  shaft  and  the 
radius  r. 

Now  it  is  not  sufficient  to  have  the  relation  between  the 
masses  and  radii  determined  by  the  formula  w\r\  +  w2rz  =  wr 
alone,  because  this  condition  only  means  that  the  shaft  will 
be  in  static  equilibrium,  or  will  be  balanced  if  the  shaft  is  sup- 
ported at  rest  on  horizontal  knife  edges.  When  the  shaft  re- 
volves, however,  there  may  be  a  tendency  for  it  to  "tilt"  in  the 
plane  containing  its  axis  and  the  radii  of  the  three  weights,  and 
this  can  only  be  avoided  by  making  the  sum  of  the  moments  of 


310  THE  THEORY  OF  MACHINES 

CO2 

the  quantities  r  X  w  X  —  about  an  axis  through  the  shaft  normal 

to  the  last-mentioned  plane,  equal  to  zero. 

For  convenience,  select  the  axis  in  the  plane  in  which  wi  re- 
volves, and  let  a  and  a2  be  the  respective  distances  of  the  planes 
of  rotation  of  w  and  Wz  from  the  axis;  then  the  moment  equation 
gives 


(wra  —  WzTzdz)  —  =  0  or  wra  = 
y 

Combining  this  relation  with  the  former  one 


gives 

/-•        a\  a 

=  wr(l  --    and  w<>r<>  =  w  r  — 

O,2/  0, 


0,2 


so  that  wtfi  and  W2r2  are  readily  determined. 

As  an  example  let  w  =  10  lb.,  r  =  2  in.,  a  =  4  in.  and  a^  =  10 


in.;  then  w2r2  =  ^f2>  an4  if  rz  be  taken  as  4  in.  w2  =          =  % 

7*2 

=  2  lb.,  since  the  radii  are  to  be  in  feet.     Further,  the  value  of 


wr 


)   =  10  X  TO!  —  T7^  —     =  1»  from  which  if 

I  l-^l  1"  I 


Q>2 

12 

be  arbitrarily  chosen  as  4  lb.,  it  will  have  to  revolve  at  a  radius 
of  y±  ft.  or  3  in.  from  the  shaft  center.  In  this  way  the  two 
weights  are  found  in  the  selected  planes  which  will  balance  the 
crankpin. 

245.  Balancing  Any  Number  of  Rotating  Masses  Located  on 
Different  Planes  Normal  to  a  Shaft  Revolving  at  Uniform 
Speed.  —  Let  there  be  any  number  of  masses,  say  four,  of  weights 
Wi,  102,  Ws  and  w±,  rotating  at  respective  radii  ri,  r2,  r3  and  r4  on  a 
shaft  with  fixed  axis  and  which  is  turning  at  w  radians  per  second, 
the  whole  being  as  shown  at  Fig.  181.  It  is  required  to  balance 
the  arrangement. 

As  before,  this  may  be  done  by  the  use  of  two  additional 
weights  revolving  with  the  shaft  and  located  in  two  planes  of 
revolution  which  may  be  arbitrarily  selected;  these  are  shown  in 
the  figure,  the  one  containing  the  point  0,  and  the  other  at  At 
and  the  quantities  ai,  a2,  as,  a*  and  a5  represent  the  distances 
of  the  several  planes  of  revolution  from  0. 


BALANCING  OF  MACHINERY 


311 


It  is  convenient  to  use  the  left-hand  plane,  or  that  through  0, 
as  the  plane  of  reference  and,  in  fact,  the  reference  plane  must 
always  contain  one  of  the  unknown  masses,  and  it  will  be  evident 

that  if  the  masses  are  balanced  the  vector  sum  -   X  r  X  co2  must 

g 

be    zero.     Further,    the    vector    sum  of   the    tilting    moments 

2 

w  X  r  X  a  X  —  of  the  various  masses  in  planes  containing  the 

masses  and  the  shaft  must  also  be  zero;  otherwise,  although  the 
system  may  be  in  equilibrium  when  at  rest,  it  will  not  be  so  while 
it  is  in  motion.  Now,  since  o>2  and  g  are  the  same  for  all  the 


D 


[  I             ] 

J 

:  i  i 

03             > 

—              a  4    - 

2                       "'                                  —  *1 

FIG.  181. — Balancing  revolving  masses. 

masses,  therefore,  the  above  equations  may  be  reduced  to  the 
form:  (1)  vector  sum  of  the  products  w  X  r  must  be  zero;  and  (2) 
vector  sum  of  the  products  wra  must  be  zero.  Since  the  first  of 
these  is  the  condition  to  be  observed  if  the  shaft  is  stationary, 
it  may  be  called  the  static  condition,  while  the  second  is  the 
dynamic  condition  coming  into  play  only  when  the  shaft  is 
revolving. 

Now  the  tilting  moment  w  X  r  X  a  has  a  tendency  to  tilt  the 
shaft  in  the  plane  containing  r  and  the  shaft,  and  it  will  be  most 
convenient  to  represent  it  by  a  vector  parallel  to  the  trace  of 


312  THE  THEORY  OF  MACHINES 

this  plane  on  the  plane  of  revolution,  or  what  is  the  same  thing, 
by  a  vector  parallel  with  the  radius  r  itself,  and  a  similar  method 
will  be  used  with  other  tilting  moments.  Two  balancing  weights 
will  be  required,  w  at  an  arbitrarily  selected  radius  r  in  the  normal 
plane  through  0,  and  w^  at  a  selected  radius  r$  in  the  normal 
plane  through  A. 

Now  from  the  static  condition  the  vector  sum 

wr  +  WiTi  +  wtfz  +  wsr3  +  w^r*  +  w^r^  =  0 


where  w  and  wb  are  unknown,  and  these  cannot  yet  be  found 
because  the  directions  of  the  radii  r  and  r5  are  not  known.  Again, 
since  the  reference  plane  passes  through  0,  tilting  moments  about 
O  must  balance,  or 

w3r  3a3  +  w^r^a^  +  wbrba^  =  0 


and  here  the  only  unknown  is  w-0r^a^  which  may  therefore  be 
determined.  The  vector  polygon  for  finding  this  quantity  is 
shown  at  (a)  in  Fig.  181  and  on  dividing  by  a&  the  value 
of  iy5r5  is  given.  The  force  polygon  shown  at  (&)  may  now  be 
completed,  and  the  only  other  unknown  w  X  r  found,  and  thus 
the  magnitude  and  positions  of  the  balancing  weights  w  and  w$ 
may  be  found.  The  construction  gives  the  value  of  the  prod- 
ucts wr  and  w$r$  so  that  either  w  or  r  may  be  selected  as 
desired  and  the  remaining  factor  is  easily  computed. 

By  a  method  similar  to  the  above,  therefore,  any  number  of 
rotating  masses  in  any  positions  may  be  balanced  by  two  weights 
in  arbitrarily  selected  planes.  Many  examples  of  this  kind 
occur  in  practice,  one  of  the  most  common  being  in  locomotives 
(see  Sec.  253),  where  the  balancing  weights  must  be  placed  in  the 
driving  wheels  and  yet  the  disturbing  masses  are  in  other  planes. 

246.  Numerical  Example  on  Balancing  Revolving  Masses.  — 
Let  there  be  any  four  masses  of  weights  Wi  —  10  lb.,  wz  =  6  Ib. 
ws  =  8  lb.  and  u'4  =  12  lb.,  rotating  at  radii  r\  —  6  in.,  r^  =  8  in., 
r3  =  9  in.  and  r±  =  4  in.  in  planes  located  as  shown  on  Fig.  12.8 
It  is  required  to  balance  the  system  by  two  w'eights  in  the  planse 

through  the  points  0  and  A  respectively. 

/> 

The  data  of  the  problem  give  w\ri   =    10   X  TO  =   5,  W^TZ   = 
6  X  TO  =  4,  w3r3  =  8  X  TO  =  6  and  w^t  =    12  X  T^  =  4,  and 


BALANCING  OF  MACHINERY 


313 


further 


=  5  X  TO  =  2.5, 


"I  O 

=  4  X     o  =  3.33, 


-j  r 


=  6  X        =  7.5  and 


10 

=  4  X         =  6. 


The  first  thing  is  to  draw  the  tilting-couple  vector  polygon  as 
shown  on  the  left  of  Fig.  182  and  the  only  unknown  here  is  wbrba$ 
which  may  thus  be  found  and  scales  off  as  4.65.  Dividing  by  a& 

12 

=  -=  =  1  ft.  gives  wbr$  =  4.65  and  the  direction  of  r5  is  also  given 


as  parallel  to  the  vector 

Next  draw  the  vector  diagram  for  the  products  w.r  as  shown 
on  the  right  of  Fig.  182,  the  only  unknown  being  the  product  wr 


FIG.  182. 

for  the  plane  through  0.  From  the  polygon  this  scales  off 
as  2.9  and  the  direction  of  r  is  parallel  to  the  vector  in  the 
diagram. 

In  this  way  the  products  wr  and  w^rb  are  known  in  magnitude 
and  direction,  and  then,  on  assuming  the  radii,  the  weights  are 
easily  found.  This  has  been  done  in  the  diagram.  It  is  advis- 
able to  check  the  work  by  choosing  a  reference  plane  somewhere 
between  0  and  A  and  making  the  calculations  again. 


BALANCING  OF  NON-ROTATING  MASSES 

247.  The  Balancing  of  Reciprocating  and  Swinging  Masses. — 

The  discussion  in  the  preceding  sections  shows  that  it  is  always 


314 


THE  THEORY  OF  MACHINES 


possible  to  balance  any  number  of  rotating  masses  by  means 
of  two  properly  placed  weights  in  any  two  desired  planes  "of 
revolution,  and  the  method  of  determining  these  weights  has 
been  fully  explained.  The  present  and  following  sections  deal 
with  a  much  more  difficult  problem,  that  of  balancing  masses 
which  do  not  revolve  in  a  circle,  but  have  either  a  motion  of 
translation  at  variable  speed,  such  as  the  piston  of  an  engine  or 
else  a  swinging  motion  such  as  that  of  a  connecting  rod  or  of  the 
jaw  of  a  rock  crusher  or  other  similar  part.  Such  problems 
not  only  present  much  difficulty,  but  their  exact  solution  is 
usually  impossible  and  all  that  can  generally  be  done  is  to  parti- 
tially  balance  the  parts  and  so  minimize  the  disturbing  effects. 
248.  Balancing  Reciprocating  Parts  Having  Simple  Harmonic 
Motion. — The  first  case  considered  is  that  of  the  machine  shown 


FIG.  183. 

in  Fig.  6  somewhat  in  detail  and  a  diagrammatic  view  of  which 
is  given  in  Fig.  183.  The  crank  a  is  assumed  to  revolve  with 
uniform  angular  velocity  co  radians'  per  second,  the  piston  c  hav- 
ing reciprocating  motion,  and  it  has  been  shown  in  Sec.  234  that 
the  acceleration  of  c  is,  at  any  instant,  equal  to  the  projection  of 
a  upon  the  direction  of  c  multiplied  by  co2,  or  the  acceleration  of 
the  piston  is  OA  X  co2  =  a  cos  6  X  co2.  The  force  necessary  to 

produce  this  acceleration  of  the  piston  then  is  F  =  —  X  a  cos  6  X  co2 

y 

where  w  is  the  weight  of  the  piston,  and  it  is  this  force  F 
which  must  be  applied  to  give  a  balance.  A  little  consideration 
will  show  that  this  force  F  is  constant  in  direction,  always  co- 
inciding with  the  direction  of  motion  of  c,  but  it  is  variable  in 
magnitude,  since  the  latter  depends  on  the  crank  angle  0. 


BALANCING  OF  MACHINERY 


315 


Suppose  now  that  at  P  is  placed  a  weight  w  exactly  equal  to  the 
weight  of  the  reciprocating  parts;  then  the  centrifugal  force  act- 
ing radially  is  a  X  co2  and  the  resolved  part  of  this  in  the  direction 

of  motion  of  c  is  clearly  —  X  a  cos  6  X  o>2,  that  is  to  say,  the 

y 

horizontal  resolved  part  of  the  force  produced  by  the  weight  w  at 
P  is  the  same  as  that  due  to  the  motion  of  the  piston.  It  there- 
fore follows  that  if  at  PI,  located  on  PO  produced  so  that  OPi 
=  OP,  there  is  placed  a  concentrated  weight  of  w  lb.,  the  hori- 
zontal component  of  the  force  produced  by  it  will  balance  the 
reciprocating  masses;  the  vertical  component,  however,  of  the 
force  due  to  w  at  PI  is  still  unbalanced  and  will  cause  vibrations 
vertically.  Thus,  the  only  effect  produced  by  the  weight  w  at 
PI  is  to  change  the  horizontal  shaking  forces  due  to  c  into  vertical 
forces,  and  the  machine  still  has  the  unbalanced  vertical  forces. 


FIG.  184. 

Frequently  in  machinery  there  is  no  real  objection  to  this 
vertical  disturbing  force,  because  it  may  be  taken  up  by  the 
foundation  of  the  machine,  but  in  portable  machines,  such  as  loco- 
motives or  fire  engines  or  automobiles,  it  may  cause  trouble 
also.  It  is  seen,  however,  that  complete  balance  is  not  ob- 
tained in  this  way,  that  is,  a  single  revolving  mass  cannot  be 
made  to  balance  a  reciprocating  mass. 

There  is  only  one  way  in  which  such  a  mass  can  be  completely 
balanced  and  that  is  by  duplication  of  the  machine.  Thus,  if  it 
were  possible  to  use  PI  as  a  crank  and  place  a  second  piston,  as 
shown  in  Fig.  184,  the  masses  would  be  completely  balanced. 
If  the  second  machine  cannot  be  placed  in  the  same  plane  nor- 
mal to  the  shaft  as  the  first,  then  balance  could  be  obtained 
by  dividing  it  into  two  parts  each  having  reciprocating  weights 


316 


THE  THEORY  OF  MACHINES 


"2  and  moving  in  planes  equidistant  from  the  plane  of  the  first 

machine. 

When  the  reciprocating  mass  moves  in  such  a  way  that  its 
position  may  be  represented  by  such  a  relation  as  a  cos  9  it  is 
said  to  have  simple  harmonic  motion  and  its  acceleration  may 
always  be  represented  by  the  formula  a  cos  6  X  co2.  Balancing 
problems  connected  with  this  kind  of  motion  are  problems  in 
primary  balancing  and  are  applicable  to  cases  where  the  connect- 
ing rod  is  very  long,  giving  approximate  results  in  such  cases,  and 
exact  results  in  cases  where  the  rod  is  infinitely  long,  and  in  the 
case  shown  in  Fig.  183,  just  discussed. 

One  method  in  which  revolving  weights  may  be  used  to  produce 
exact  balance  in  the  case  of  a  part  having  simple  harmonic 


Gears 


Geared  to 

Crank  Shaft 

Ratio  1:1 

(JLof  Engine 


FIG.  185. — Engine  balancing — primary  balance. 

motion  is  shown  in  Fig.  185,  where  the  two  weights  %w  are 
equal  and  revolve  at  the  speed  of  the  crank  and  in  opposite 
sense  to  one  another,  their  combined  weight  being  equal  to  the 
weight  w  of  the  reciprocating  parts.  Evidently  here  the  vertical 
components  of  the  two  weights  balance  one  another,  leaving  their 
horizontal  components  free  to  balance  the  reciprocating  parts. 
Taking  the  combined  effective  weights  as  equal  to  that  of  the 
reciprocating  parts,  then  they  must  rotate  at  a  radius  equal  to 
that  of  the  crank,  and  must  be  180°  from  the  latter  when  it  is 
on  the  dead  center. 

249.  Reciprocating  Parts  Operated  by  Short  Connecting  Rod. 
— The  general  construction  adopted  in  practice  for  moving 
reciprocating  parts  differs  from  Fig.  183  in  that  the  rod  imparting 


BALANCING  OF  MACHINERY 


317 


the  motion  is  not  so  long  that  the  parts  move  with  simple  har- 
monic motion,  and  in  the  usual  proportions  adopted  in  engines 
the  variation  is  quite  marked,  for  the  rods  are  never  longer  than 
six  times  the  crank  radius  and  are  often  as  small  as  four  and  one- 
half  times  this  radius. 

The  method  to  be  adopted  in  such  cases  is  to  determine  the 
acceleration  of  the  reciprocating  parts  and  to  plot  it  for  each  of 
the  crank  angles  as  described  in  the  preceding  chapter.  To 
illustrate  this,  suppose  it  is  required  to  balance  the  reciprocating 
parts  in  the  engine  examined  in  Sec.  241 ;  then  the  accelerations 
of  these  are  found  and  set  down  as  shown  in  the  table  belonging 
to  this  case.  The  accelerations  shown  in  the  third  column  have 


1000 
em 

^ 

^ 

—  , 

X 

7 

^ 

Accelerations  -Ft.  per  Sect  per  Sec, 

iiiigoiii* 

\j 

\ 

^ 

t 

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x\ 

C 

, 

-*£ 

x 

\\ 

0 

^ 

•*" 

•-^ 

// 

^ 

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8      5 

G     'i 

1^ 

\    10^    ^ 

2G    1 

14    1( 

>2    IS 

0    1 

38   2 

6    £ 

4^2. 

2 

7A 

®*3Qb    324    342  3(X 
Crank  lAngles 

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\ 

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^ 

X 

^ 

— 

A 

—  .     -* 

~B~ 



FIG.  186. 

been  plotted  in  the  plain  line  A  on  Fig.  186.  This  curve  must 
now  be  broken  up  into  its  corresponding  harmonic  components, 
and  it  is  usual  to  assume  that  these  are  in  phase  at  the  inner 
dead  center  with  the  original  curve.  The  dotted  line  B  repre- 
sents a  simple  harmonic  or  sine  curve  in  phase  with  the  plain 
curve  and  having  maximum  height  of  J£(  1,053  +  711)  =  882, 
which  is  the  mean  value  for  the  true  curve  heights  at  0°  and 
at  180°  crank  angles.  The  difference  between  these  two  curves 
has  been  plotted  in  the  broken  line  curve  C  and  will  be  found  on 
examination  to  be  almost  a  true  sine  curve,  in  fact,  it  differs 
so  little  from  a  sine  curve  that  it  would  be  impossible  to  distin- 
guish between  them  on  the  scale  of  this  drawing.1 

It  will  be  observed  that  the  curve  C  is  also  in  phase  at  the 

1  See  Appendix  A  for  mathematical  proof  of  these  statements. 


318  THE  THEORY  OF  MACHINES 

inner  dead  center  with  the  curve  A  but  has  twice  the  frequency 
and  maximum  height  on  the  drawing  of  171  ft.  per  second  per 

second.     It  will  also  be  found' that  171  is  ~  X  882  or  -—•  X  882 

0  lo 

=  171. 

The  reciprocating  parts  of  this  engine  could  also  be  balanced 
in  the  manner  shown  in  Fig.  185,  but  it  would  require  that  in- 
stead of  one  pair  of  weights,  two  pairs  should  be  used;  one  pair 
rotating  at  the  speed  of  the  crank  and  180°  from  it  at  the  dead 
centers,  and  another  pair  in  phase  at  the  inner  dead  center  with 
the  first  but  rotating  at  double  the  speed  of  the  crank.  The 
weight  rotating  at  the  speed  of  the  crankshaft  should  be  the 
same  as  that  of  the  piston,  namely  161  Ib.  (m  =  5),  if  placed  at 
at  3^2  in.  radius,  while  the  weight  making  twice  the  speed  (co 
=  110)  of  the  crank  might  also  be  placed  at  a  radius  of  3^  m-> 

.  ,     5  X  171  X  32.2 
in  which  case  it  would  weigh      Q  -  -  =  7.76  Ib.     In 

jj  X  (110)' 

order  that  these  weights  could  rotate  without  interference  they 
might  have  to  be  divided  and  separated  axially,  in  which  case 
the  two  halves  of  the  same  weight  would  have  to  be  placed  equi- 
distant from  the  plane  of  motion  of  the  connecting  rod. 

It  is  needless  to  say  that  the  arrangement  sketched  above  is 
too  complicated  to  be  used  to  any  extent  except  in  the  most 
urgent  cases,  where  some  serious  disturbance  results.  Counter- 
weights attached  directly  to  the  crankshaft  are  sometimes  used, 
but  at  best  these  can  only  balance  the  forces  corresponding  to 
the  curve  B  and  always  produce  a  lifting  effect  on  the  engine. 
The  reader  must  note  that  the  above  method  takes  no  account 
of  the  weight  of  the  connecting  rod,  which  will  be  considered  later. 

If  the  method  already  described  cannot  be  used,  then  the  only 
other  method  is  by  duplication  of  the  parts  and  this  will  be  de- 
scribed at  a  later  stage. 

Where  the  acceleration  of  the  reciprocating  masses  cannot  be 
represented  by  a  simple  harmonic  curve,  but  must  have  a  second 
harmonic  of  twice  the  frequency,  superposed  on  it,  the  problem 
is  one  of  secondary  balancing,  so  called  because  of  the  latter 
harmonic. 

250.  Balancing  Masses  Having  Any  General  Form  of  Plane 
Motion. — It  is  impossible,  in  general,  to  balance  masses  moving 
in  a  more  or  less  irregular  way,  such,  for  example,  as  the  jaw  of 


BALANCING  OF  MACHINERY  319 

the  rock  crusher  shown  in  Fig.  168,  or  the  connecting  rod  of  an 
engine.  The  general  method,  however,  is  to  plot  the  curve  of 
accelerations  for  the  center  of  gravity  of  the  mass,  using  crank 
angles  as  a  base,  and  if  the  resulting  curve  at  all  approximates 
to  a  simple  harmonic  curve  a  weight  may  be  attached  to  the 
crank,  as  already  described,  which  will  roughly  balance  the 
forces,  or  will  at  least  reduce  them  very  greatly.  The  magnitude 
of  the  weight  and  its  position  will  be  found  by  drawing  a  simple 
harmonic,  curve  which  approaches  most  nearly  to  the  actual 
acceleration  curve.  As  the  accelerations  are  not  all  in  the  same 
direction,  the  most  correct  way  is  to  plot  two  curves  giving  the 
resolved  parts  of  the  acceleration  in  two  planes  and  balance  each 
separately,  but  usually  an  approximate  result  is  all  that  is  desired, 
and,  as  the  shaking  forces  are  mainly  in  one  plane,  the  resolved 
part  in  this  plane  alone  is  all  that  is  usually  balanced. 

In  engines,  the  method  of  balancing  the  connecting  rod  is 
somewhat  different  to  that  outlined  above.  The  usual  plan  is 
to  divide  the  rod  up  into  two  equivalent  masses  in  the  manner 
described  in  Sec.  238,  one  of  the  masses  mi  being  assumed  as 
located  at  the  wristpin  and  the  location  of  the  other  mass  mz 
is  found  as  described  in  the  section  referred  to.  In  this  way  the 
one  mass  mi  may  be  regarded  simply  as  an  addition  to  the  weight 
of  the  reciprocating  masses  and  balancing  of  it  effected  as  de- 
scribed in  the  last  or  following  sections.  The  other  mass  ra2, 
however,  gives  trouble  and  cannot,  as  a  matter  of  fact,  be  exactly 
balanced  at  all,  so  that  there  is  still  an  unbalanced  mass. 

Consideration  of  a  number  of  practical  cases  shows  that  w2 
in  many  steam  engines  lies  close  to  the  center  of  the  crankpin. 
The  engine  discussed  in  Sec.  241,  for  instance,  has  the  mass  mi 
concentrated  at  the  wristpin  and  the  mass  m2  will  be  only  0.36  in. 
away  from  the  crankpin;  for  long-stroke  engines,  however,  the 
mass  m2  may  be  some  distance  from  the  crankpin  and  in  such 
cases  the  method  described  below  will  not  give  good  results.  In 
automobile  engines  the  usual  practice  is  to  make  the  crank  end 
of  the  rod  very  much  heavier  and  the  crankpin  larger  than  the 
same  quantities  at  the  piston  end  and  hence  the  first  statement 
of  this  paragraph  is  not  true.  In  one  rod  examined  the  length 
between  centers  was  12  in.,  and  the  center  of  gravity  3.03  in. 
from  the  crankpin  center;  the  weight  of  the  rod  was  2.28  Ib.  and 
selecting  the  mass  mi  at  the  wristpin  its  weight  would  be  0.43  Ib. 
and  the  remaining  weight  would  be  1.85  Ib.  concentrated  0.92  in. 


320  THE  THEORY  OF  MACHINES 

from  the  crankpin  and  on  the  wristpin  side  of  it,  so  that  consider- 
able error  might  result  by  assuming  the  latter  mass  at  the  crank- 
pin  center. 

The  fact  that  the  mass  w2  does  not  fall  exactly  at  the  crankpin 
has  been  already  explained  in  Sec.  241,  and  in  the  engine  there 
discussed  the  resultant  force  on  the  rod  passes  through  L,  slightly 
to  the  right  of  the  crankshaft,  instead  of  passing  through  this 
center,  as  it  would  do  if  the  mass  w2  fell  at  the  crankpin.  If  an 
approximation  is  to  be  used,  and  it  appears  to  be  the  only  thing 
to  do  under  existing  conditions,  m2  may  be  assumed  to  lie  at  the 
crankpin,  and  thus  the  rod  is  divided  into  two  masses;  one,  mi 
concentrated  at  the  wristpin  and  balanced  along  with  the  recip- 
rocating masses,  and  the  other,  m2,  concentrated  at  the  crank- 
pin  and  balanced  along  with  the  rotating  masses. 

It  should  be  pointed  out  in  passing,  that  the  method  of  divid- 
ing the  rod  according  to  the  first  of  two  equations  of  Sec.  238, 
that  is,  so  that  their  combined  center  of  gravity  lies  at  the  true 
center  of  gravity  of  the  rod,  to  the  neglect  of  the  third  equation, 
leads  to  errors  in  some  rods.  Much  more  reliable  results  are 
obtained  by  finding  wii  and  w2  according  to  the  three  equations  in 
Sec.  238  and  the  examples  of  Sec.  241,  except  that  m2  is  assumed 
to  be  at  the  crankpin  center.  Dividing  .the  mass  m  so  that  its 
components  mi  and  w2  have  their  center  of  gravity  coinciding 
with  that  of  the  actual  rod  will  usually  give  fairly  good  results,  if 
the  diameters  of  the  crankpin  and  wristpin  do  not  differ  unduly. 

251.  Balancing  Reciprocating  Masses  by  Duplication  of  Parts. 
— Owing  to  the  complex  construction  involved  when  the  recipro- 
cating masses  are  balanced  by  rotating  weights,  such  a  plan  is 
rarely  used,  the  more  common  method  being  to  balance  the  recip- 
rocating masses  by  other  reciprocating  masses.  The  method 
may  best  be  illustrated  in  its  application  to  engines,  and  indeed 
this  is  where  it  finds  most  common  use,  automobile  engines  being 
a  notable  example. 

For  a  single-cylinder  engine  the  disturbing  forces  due  to  the 
reciprocating  masses  are  proportional  to  the  ordinates  to  such 
a  curve  as  A,  Fig.  186,  or  what  is  the  same  thing  to  the  sum  of 
the  ordinates  to  the  curves  B  and  (7.  Suppose  now  a  second  en- 
gine, an  exact  duplicate  of  the  first,  was  attached  to  the  same 
shaft  as  the  former  engine  and  let  the  cranks  be  set  180°  apart 
as  at  Fig.  187 (a),  then  it  is  at  once  evident  that  there  will  be  a 
tilting  moment  normal  to  the  shaft  in  the  plane  passing  through 


BALANCING  OF  MACHINERY 


321 


the  axis  of  the  shaft  and  containing  the  reciprocating  masses, 
and  further,  a  study  of  Fig.  186  will  show  that  while  the  ordinates 
to  the  two  curves  B  belonging  to  these  machines  neutralize,  still 
the  two  curves  C  are  additive  and  there  is  unbalancing  due  to  the 
forces  corresponding  to  curves  C.  In  Fig.  187  are  shown  at  (6) 
and  (c)  two  other  arrangements  of  two  engines,  both  of  which 
eliminate  the  tilting  moments;  in  the  arrangement  (6)  the  cranks 
are  at  180°  and  the  unbalanced  forces  are  completely  eliminated, 
producing  perfect  balance,  whereas  at  (c)  the  sum  of  the  crank 
angles  for  the  two  opposing  engines  is  180°  and  the  forces  cor- 
responding to  curve  B  are  balanced,  while  those  corresponding 
to  C  are  again  unbalanced  and  additive,  so  that  there  is  still  an 
unbalanced  force.1  Since  the  disturbing  forces  are  in  the  direc- 


O)  CO 

FIG.  187. — Different  arrangements  of  engines. 

tion  of  motion  of  the  pistons,  nothing  would  be  gained  in  this 
respect  by  making  the  cylinder  directions  different  in  the  two 
cases. 

If  a  three-cylinder  engine  is  made,  with  the  cylinders  side  by 
side  and  cranks  set  at  120°  as  in  Fig.  188,  an  examination  by  the 
aid  of  Fig.  186  will  show  that  the  arrangement  gives  approxi- 
mately complete  balance,  since  the  sum  of  ordinates  to  the  curves 
B  and  C  for  the  three  will  always  be  zero,  but  there  is  still  a 
tilting  moment  normal  to  the  axis  of  the  crankshaft  which  is 
unavoidable.  Four  cylinders  side  by  side  on  the  same  shaft, 
with  the  two  outside  cranks  set  together  and  the  two  inner  ones 
also  together  and  set  180°  from  the  other,  does  away  with  the 
tilting  moment  of  Fig.  187  (a)  but  still  leaves  unbalanced  forces 
proportional  to  the  ordinates  to  the  curves  C.  A  six-cylinder 
arrangement  with  cylinders  set  side  by  side  and  made  of  two 

1  In  order  to  get  a  clear  grasp  of  these  ideas  the  reader  is  advised  to  make 
several  separate  tracings  of  the  curves  B  and  C,  Fig.  186,  and  to  shift  these 
along  relatively  to  one  another  so  as  to  see  for  himself  that  the  statements 
made  are  correct. 
21 


322  THE  THEORY  OF  MACHINES 

parts  exactly  like  Fig.  188,  but  with  the  two  center  cranks  parallel 
gives  complete  balance  with  the  approximations  used  here. 

In  what  has  been  stated  above  the  reader  must  be  careful  to 
remember  that  the  rod  has  been  divided  into  two  equivalent 
masses  and  the  discussion  deals  only  with  the  balancing  of  the 
reciprocating  part  of  the  rod  and  the  other  reciprocating  masses. 
The  part  acting  with  the  rotating  masses  must  also  be  balanced, 
usually  by  the  use  of  a  balancing  weight  or  weights  on  the  crank- 
shaft according  to  the  method  already  described  in  Sec.  245.  It 
is  to  be  further  understood  that  certain  approximations  have  been 
introduced  with  regard  to  the  division  of  the  connecting  rod,  and 
also  with  regard  to  the  breaking  up  of  the  actual  acceleration 
curve  for  the  reciprocating  masses  into  two  simple  harmonic 
curves,  one  having  twice  the  frequency  of  the  other.  Such  a 
division  is  a  fairly  close  approximation,  but  is  not  exact. 


FIG.  188. 

The  shape  of  the  acceleration  curves  A,  Fig.  186,  and  its  com- 
ponents B  and  C,  depend  only  upon  the  ratio  of  the  crank  radius 
to  the  connecting-rod  length,  and  also  upon  the  angular  velocity 

co.     For  the  same  value  of  T  the  curves  will  have  the  same  shape 

for  all  engines,  and  the  acceleration  scale  can  always  be  readily 
determined  by  remembering  that  at  crank  angle  zero  the  accelera- 
tion is  f a  +  -j- }  co2  ft.  per  second  per  second.  These  curves  also 

represent  the  tilting  moment  to  a  certain  scale  since  the  mo- 
ment is  the  accelerating  force  multiplied  by  the  constant  distance 
from  the  reference  plane. 

The  chapter  will  be  concluded  by  working  out  a  few  practical 
examples. 

252.  Determination  of  Crank  Angles  for  Balancing  a  Four- 
Cylinder  Engine. — An  engine  with  four  cylinders  side  by  side 
and  of  equal  stroke,  is  to  have  the  reciprocating  parts  balanced  by 
setting  the  crank  angles  and  adjusting  the  weights  of  one  of  the 
pistons.  It  is  required  to  find  the  proper  setting  and  weight, 
motion  of  the  piston  being  assumed  simple  harmonic.  The 


BALANCING  OF  MACHINERY 


323 


dimensions  of  the  engine  and  all  the  reciprocating  parts  but  one 
set  are  given. 

Let  Fig.  189  represent  the  crankshaft  and  let  w2,  WB,  w4  repre- 
sent the  known  weights  of  three  of  the  pistons,  etc.,  together 
with  the  part  of  the  connecting  rod  taken  to  act  with  each  of 
them  as  found  in  Sec.  250.  It  is  required  to  find  the  remain- 
ing weight  Wi  and  the  crank  angles. 

Choose  the  reference  plane  through  Wi,  and  all  values  of  r  are 
the  same;  also  the  weights  may  be  transferred  to  the  respective 
crankpins,  Sec.  248,  as  harmonic  motion  is  assumed.  Draw  the 
wra  triangle  with  sides  of  lengths  W2ra2)  W3ra3  and  W4ra4  which 
gives  the  directions  of  the  three  cranks  2,  3,  and  4.  Next  draw 
the  wr  polygon,  from  which  WIT  is  found,  and  thus  Wi,  and  the 


700 
Moments    wra  Forces   wr 

FIG.  189. — Balancing  a  four  crank  engine. 

corresponding  crank  angle.  The  part  of  the  rods  acting  at  the 
respective  crankpins,  as  well  as  the  weight  of  the  latter,  must  be 
balanced  by  weights  determined  as  in  Sec.  245.  The  four  recip- 
rocating weights  operated  by  cranks  set  at  the  angles  found 
will  be  balanced,  however,  if  harmonic  motion  is  assumed. 

Example. — Let  w2  =  250  lb.,  wz  =  220  lb.  and  iv*  '=  200  lb., 
r  =  6  in.  and  the  distance  between  cylinders  as  shown.  Then 
WzTz  =  125,  WsTa  =  110,  i#4r4  =  100,  WzTtfig  =  250,  w^fya^  = 
550  and  i(;4r4a4  =  700.  The  solution  is  shown  on  Fig.  189 
which  gives  the  crank  angles  and  weight  Wi  =  216  lb.  The 
rotating  weights  would  have  to  be  independently  balanced. 

253.  Balancing  of  Locomotives. — In  two-cylinder  locomotives 
the  cranks  are  at  90°,  and  the  balance  weights  must  be  in  the 


324 


THE  THEORY  OF  MACHINES 


driving  wheels.  In  order  to  avoid  undue  vertical  forces  it  is 
usual  to  balance  only  a  part  of  the  reciprocating  masses,  usually 
about  two-thirds,  by  means  of  weights  in  the  driving  wheels, 
and  these  balancing  weights  are  also  so  placed  as  to  compensate 
for  the  weights  of  the  cranks.  Treating  the  motion  of  the  piston 
as  simple  harmonic,  this  problem  gives  no  difficulty. 

Example. — Let  a  locomotive  be  proportioned  as  shown  on  Fig. 
190.  The  piston  stroke  is  2  ft.  and  the  weight  of  the  revolving 
masses  is  equivalent  to  620  Ib.  attached  to  the  crankpin.  The 
reciprocating  masses  are  assumed  to  have  harmonic  motion  and 
to  weigh  550  Ib.  and  only  60  per  cent,  of  these  latter  masses  are 
to  be  balanced,  so  that  weight  at  the  crankpin  corresponding 
to  both  of  these  will  be  620  +  0.60  X  550  =  950  Ib.  for  each  side. 


Moments    WrcC 
5740 


1                  1 

1 

i  3 

T 

I 

3 

11    , 

FIG.  190. — Locomotive  balancing. 


The  reference  plane  for  the  tilting  moments  must  always  pass 
through  one  of  the  unknown  masses,  and  the  plane  is  here  taken 
through  the  wheel  2.  Note  that  the  crank  1  being  on  opposite 
side  of  the  reference  plane  to  wheel  3  and  crank  4,  the  sense  of 
the  moment  vector  must  be  opposite  to  what  it  would  be  if  it 
were  in  the  position  4.  The  crank  1  is  thus  drawn  from  the 
shaft  in  opposite  sense  to  the  vector  Wiriai. 

A  diagrammatic  plan  of  the  locomotive  in  Fig.  190  gives  the 
moment  arms  of  the  masses  and  the  values  of  the  corresponding 
moments  are  plotted  on  the  right.  Thus  Wir^i  =  950  X  1  X  1.12 
=  1,069  and  w  4^0,4,  =  950  X  1  X  6.04  =  5,740  and  from  the 
vector  diagram  W3r3a3  scales  off  as  5,840.  Since  a*  =  4.92  ft., 
w3r3  =  1,187.  The  force  polygon  may  now  be  drawn  with 
sides  950,  950  and  1,187  parallel  to  the  moment  vectors  and  then 
wtfz  is  scaled  off  as  1,187.  Selecting  suitable  radii  r2  and  r3 
give  the  weights  w2  and  wz  and  the  end  view  of  the  wheels  and 


BALANCING  OF  MACHINERY  325 

axle  on  the  right  shows  how  the  weights  would  be  placed  in 
accordance  with  the  results. 

The  above  treatment  deals  only  with  primary  disturbing  forces, 
only  part  of  which  are  balanced,  and  further,  it  is  to  be  noticed 
that  there  will  be  considerable  variation  in  rail  pressure,  which 
might,  with  some  designs,  lift  the  wheel  slightly  from  the  tracks 
at  each  revolution,  a  very  bad  condition  where  it  occurs. 

254.  Engines  Used  for  Motor  Cycles  and  Other  Work. — In 
recent  years  engines  have  been  constructed  having  more  than 
one  cylinder,  with  the  axes  of  all  the  cylinders  in  one  plane  nor- 
mal to  the  crankshaft.  Frequently,  in  such  engines,  all  the 
connecting  rods  are  attached  to  a  single  crankpin,  and  any 
number  of  cylinders  may  be  used,  although  with  more  than  five, 
or  seven  cylinders  at  the  outside,  there  is  generally  difficulty  in 
making  the  actual  construction.  The  example,  shown  in  Fig. 


FIG.  191. — Motor  cycle  engine. 

191,  represents  a  two-cylinder  engine  with  lines  90°  apart  and  is 
a  construction  often  used  in  motorcycles,  in  which  case  a  vertical 
line  passes  upward  through  0,  midway  between  the  cylinders; 
in  these  motorcycle  engines  the  angle  between  the  cylinders  is 
frequently  less  than  90°.  The  same  setting  has  been  in  use  for 
many  years  with  steam  engines  of  large  size,  in  which  case  one 
of  the  cylinders  is  vertical,  the  other  horizontal. 

When  a  similar  construction  is  used  for  more  than  two  cylin- 
ders, the  latter  are  usually  evenly  spaced;  thus  with  three  cylin- 
ders the  angle  between  them  is  120°. 

These  constructions  introduce  a  number  of  difficult  problems 
in  balancing,  which  can  only  be  touched  on  here,  and  the  method 
of  treatment  discussed.  The  motorcycle  engine  of  Fig.  191 


326  THE  THEORY  OF  MACHINES 

with  cylinders  at  90°  will  alone  be  considered,  and  it  will  be 
assumed  that  both  sets  of  moving  parts  are  identical  and  that 
the  weight  of  each  piston  together  with  the  part  of  the  rod  that 
may  be  treated  as  a  reciprocating  mass  is  w  Ib.  In  the  following 
discussion  only  the  reciprocating  masses  are  considered,  and  the 
part  of  the  rods  that  may  be  treated  as  masses  rotating  with  the 
crankpin,  and  also  the  crankpin  and  shaft  are  balanced  indepen- 
dently ;  as  the  determination  of  the  latter  balance  weights  offers 
no  difficulty  the  matter  is  not  taken  into  account. 

It  has  already  been  shown  in  Sec.  249  (and  in  Appendix  A)  that 
the  acceleration  of  the  piston  may  be  represented  by  the  sum  of 
two  harmonic  curves,  one  of  the  frequency  of  the  crank  rotation, 
and  another  one  of  twice  this  frequency;  these  are  shown  in  Fig. 
186  in  the  curves  B  and  C.  It  is  further  explained  in  Sec.  249 

that  the  maximum  ordinate  to  the  curve  C  is  7  times  that  to  the 

o 

curve  B. 

The  discussion  of  Sees.  248  and  249  should  also  make  it  clear 
that  if  a  weight  w  be  secured  at  S,  in  Fig.  191,  the  component  of 
the  force  produced  by  this  weight  in  the  direction  OX  will  balance 
the  primary  component  of  the  acceleration  of  the  piston  Q,  that 
is,  it  will  balance  the  accelerations  corresponding  to  the  ordinates 
to  the  curve  B,  Fig.  186.  There  is  still  unbalanced  the  accelera- 
tions corresponding  to  the  curve  C  and  also  the  vertical  compo- 
nents of  the  force  produced  by  the  revolving  weight  w  at  S,  these 
latter  being  in  the  direction  OY.  A  very  little  consideration  will 
show  that  the  latter  forces  are  exactly  balanced  by  the  recipro- 
cating mass  at  R,  or  that  the  weight  w  at  S  produces  complete 
primary  balance. 

The  forces  due  to  the  ordinates  to  the  curve  C  for  the  piston 
Q  could  be  balanced  by  another  weight,  in  phase  with  S  when  0  is 
zero,  but  rotating  at  double  the  angular  speed  of  OS.  If  this 

weight  is  at  the  crank  radius,  its  magnitude  should  be  (^j  X  r 
times  w,  since  its  angular  speed  is  double  that  of  OS,  and  also  the 
maximum  height  of  the  curve  C  is  r  times  that  of  B,  Sec.  249. 

The  difficulty  is  that  this  latter  weight  has  a  component  in  the  di- 
rection of  OF  which  will  not  be  balanced  by  the  forces  corres- 
ponding to  the  curve  C  for  the  piston  R.  It  is  not  hard  to  see 
this  latter  point,  for  when  0  becomes  90°  the  fast  running  weight 


BALANCING  OF  MACHINERY  327 

should  coincide  with  S  to  balance  the  forces  of  the  piston  R, 
whereas  if  it  coincided  with  w  when  0  is  zero  it  will  be  exactly 
opposite  to  it  when  6  =  90°. 

With  such  an  arrangement  as  that  shown  there  is,  then,  per- 
fect primary  balance  but  the  secondary  balance  cannot  be  made 
at  the  same  time.  If  the  secondary  balance  weight  coincides 
with  w  when  6  is  zero,  then  the  unbalanced  force  in  the  direction 
OF  will  be  a  maximum,  and  if  Qis  exactly  balanced  the  maximum 
unbalanced  force  in  the  direction  OF  is  twice  that  corresponding 

to  the  ordinates  to  the  curve  C,  or  is  2  X  —  X  7  X  aco2.     Owing 

Q       o 

to  the  difficult  construction  involved  in  putting  in  the  secondary 
balance  weight,  the  latter  is  not  used,  and  then  the  maximum  un- 
balanced force  may  readily  be  shown  to  be  \2  X  —  X  r-  X  aco2  pds. 

The  use  of  the  curves  like  Fig.  186  will  enable  the  reader 
to  prove  the  correctness  of  the  above  results  without  difficulty. 

In  dealing  with  all  engines  of  this  type,  no  matter  what  the 
number  or  distribution  of  the  cylinders,  the  primary  and  second- 
ary revolving  masses  are  always  to  be  found,  and  by  combining 
each  of  these  separately  for  all  the  cylinders  the  primary  and 
secondary  disturbing  forces  may  be  found,  and  the  former 
always  balanced  by  a  revolving  weight  on  the  crank,  but  the  lat- 
ter can  be  balanced  only  in  some  cases  where  it  is  possible  to 
make  the  reciprocating  masses  balance  one  another.  Thus,  a 
six-cylinder  engine  of  this  type  with  cylinders  at  equal  angles 
may  be  shown  to  be  in  perfect  balance. 

QUESTIONS  ON  CHAPTER  XVI 

1.  What  are  the  main  causes  of  vibrations  in  rotating  and  moving  parts  of 
machinery?     What  is  meant  by  balancing? 

2.  The  chapter  deals  only  with  the  balancing  of  forces  due  to  the  masses; 
why  are  not  the  fluid  pressures  considered? 

3.  If  an  engine  ran  at  the  same  speed  would  there  be  any  different  arrange- 
ment for  balancing  whether  the  crank  was  rotated  by  a  motor  or  operated  by 
steam  or  gas?     Why? 

4.  Two  masses  weighing  12  and  18  Ib.  at  radii  of  20  and  24  in.  and  inclined 
at  90°  to  one  another  revolve  in  the  same  plane.     Find  the  position  arid  size 
of  the  single  balancing  weight. 

5.  If  the  weights  in  question  4  revolve  in  planes  10  in.  apart,  find  the 
weights  in  two  other  planes  15  and  12  in.  outside  the  former  weights,  which 
will  balance  them. 


328  THE  THEORY  OF  MACHINES 

6.  Examine  the  case  of  V-type  of  engine  similar  to  Fig.  191  with  the 
cylinders  at  60°  and  see  if  there  are  unbalanced  forces  and  how  much. 

7.  A  gas  engine  15  in.  stroke  has  two  flywheels  with  the  crank  between 
them,  one  being  18  in.  and  the  other  24  in.  from  the  crank.     The  equivalent 
rotating  weight  is  200  Ib.  at  the  crankpin,  while  the  reciprocating  weight  is 
250  Ib.     Find  the  weights  required  in  the  flywheels  to  balance  all  of  these 
masses. 

8.  In  a  four-crank  engine  the  cylinders  are  all  equally  spaced  and  the 
reciprocating  weights  for  three  of  the  engines  are  300,  400  and  500  Ib.     Find 
the  weight  of  the  fourth  set  and  the  crank  angles  for  balance. 


APPENDIX  A 

Approximate  Analytical  Method  of  Computing  the  Acceleration  of  the 
Piston  of  an  Engine 

The  graphical  solution  of  the  same  problem  is  given  in  Sec.  236. 

Let  Fig.  192  represent  the  engine  and  let  0  be  the  crank  angle 
reckoned  from  the  head-end  dead  center,  and  further  let  x  denote 
the  displacement  of  the  piston  corresponding  to  the  motion  of  the 
crank  through  angle  6.  Taking  o>  to  represent  the  angular  ve- 
locity of  the  crank,  and  t  the  time  required  to  pass  through  the 
angle  0,  then  0  =  ut. 


FIG.  192. 


When  the  crank  is  in  the  position  shown,  the  velocity  of  the 

piston  is  —  and  its  acceleration  is  -j-. 
at  dt 

Examination  of  the  figure  shows  that: 


or 


x  =  a  +  b  —  a  cos  0  —  b  cos  <f> 
•x  •=  a  cos  0  +  b  cos"0  —  (b  +  a). 


/ —  la2  & 

Now  cos0  =  \1  —  sin2  <£  =\/l  —  :rr  sin20;  since  sin  <f>  =  -  sin  6. 

\          b2  b 

a2 
Further,  since  —  sin2  0  is  generally  small  compared  with  unity 


the  value  of 


B  is  equal  1  -•  p^  sin2  0  approximately. 


(It  is  in  making  this  assumption  that  the  approximation  is  intro- 
duced and  for  most  cases  the  error  is  not  serious.) 


Thus 


— x  =  a  cos 


~i sin2  e]  - 


329 


330  THE  THEORY  OF  MACHINES 


or  —  x  =  a  cos  ut  —  —  sin2  ut  —  a 

dx  a2 

therefore  —  —  =  aco  sin  co£  —  —  co  sin  at  cos 
d/  6 


=  aco  sin  co£  —  — -  co  sin  2co£ 

d2z  •    a2 

and          —  —  —  =  aco2  cos  co£  —  —  co2  cos  2co£ 
d/2  b 

a2 

=  aco-  cos  0  —  —  co2  cos  20. 
6 

Therefore  the  acceleration  is 

a2 
—f  =  aco2  cos  0  —  -j-  co2  cos  20 


=  aco 


I  cos  0  —  T  cos  201  . 


Since  x  is  negative  and  the  acceleration  is  also  negative,  the 
latter  is  toward  the  crankshaft,  in  the  same  sense  as  x. 

The  above  expression  will  be  found  to  be  exactly  correct  at  the 
two  dead  centers  and  nearly  correct  ,at  other  points.  It  shows 
that  the  acceleration  curve  for  the  piston  is  composed  of  two 
simple  harmonic  curves  starting  in  phase,  the  latter  of  which  has 

twice  the  frequency  of  the  other,  and  an  amplitude  of  y  times  the 

former's  value.  This  has  been  found  to  be  the  case  in  the  curves 
plotted  from  the  table  at  the  end  of  Chapter  XV  and  shown  on 
Fig.  186,  and  the  error  due  to  the  factor  neglected  is  found  very 
small  in  this  case. 

In  the  case  shown  in  Fig.  186,  co  is  55  radians  per  second  and 

Ql/ 

a  =  -^~  =  0.292  ft.,  so  that  the  value  of  aco2  cos  0  at  crank  angle 
iz 

0  .  =  0  is  aco2  =  0.292  X  (55)  2  =  882  ft.  per  second  per  second, 
and  this  is  the  maximum  height  of  the  first  curve.  At  the  same 


angle  0  =  0  the  value  of  aco2  X  g  cos  20  =  882  X  ~       =  171  ft. 

per  second  per  second,  which  gives  the  maximum  height  of  the 
curve  of  double  frequency.  These  values  are  the  same  as  those 
scaled  from  Fig.  186. 


APPENDIX  B 

Experimental  Method  of  Finding  the  Moment  of  Inertia  of 
Any  Body 

For  the  convenience  of  those  using  this  book  the  experimental 
method  of  finding  the  moment  of  inertia  and  radius  of  gyration 
of  a  body  about  its  center  of  gravity  is  given  herewith. 

Suppose  it  is  desired  to  find  these  quantities  for  the  connecting 
rod  shown  in  Fig.  193.  Take  the  plane  of  the  paper  as  the  plane 
of  motion  of  the  rod.  Balance  the  rod  carefully  across  a  knife 
edge  placed  parallel  with  the  plane  of  motion  of  the  rod,  and 
the  center  of  gravity  G  will  be  directly  above  the  knife  edge. 


FIG.  193. — Inertia  of  rod. 

Next  secure  a  knife  edge  in  a  wall  or  other  support  so  that  its 
edge  is  exactly  horizontal  and  hang  the  rod  on  it  with  the  knife 
edge  through  one  of  the  pin  holes  and  let  it  swing  freely  like  a 
pendulum.  By  means  of  a  stop  watch  find  exactly  the  time 
required  to  swing  from  one  extreme  position  to  the  other;  this 
can  be  most  accurately  found  by  taking  the  time  required  to  do 
this  say,  100  times.  Let  t  sec.  be  the  time  for  the  complete 
swing. 

Next  measure  the  distance  h  feet  from  the  knife  edge  to  the 
center  of  gravity,  and  also  weigh  the  rod  and  get  its  exact  weight 
w  Ib. 

Then  it  is  shown  in  books  on  mechanics  that 

/2  ni\ 

'-^X^-gO**' 

in  foot  and  pound  units,  gives  the  moment  of  inertia  of  the  rod 
abtout  its  center  of  gravity. 

331 


332  THE  THEORY  OF  MACHINES 

As  an  example,  an  experiment  was  made  on  an  automobile 
rod  of  12-in.  centers  and  weighing  2  Ib.  4>^  oz.  or  2.281  Ib.  The 
crank  and  wristpins  were  respectively  2  in.  and  !^{Q  in.  diameter, 
and  when  placed  sideways  on  a  knife  edge  it  was  found  to  bal- 
ance at  a  point  3.03  in.  from  the  center  of  the  crankpin.  The 
rod  was  first  hung  on  a  knife  edge  projecting  through  the  crank- 
pin  end,  so  that  h  =  4.03  in.  or  0.336  ft.,  and  it  was  found  that 
it  took  94%  sec.  to  make  200  swings;  thus 

94  6 
t  =  =  0.473  sec. 


(0  473V  2  281 

Then  I  =  '^  X  2.281  X  0.336  -  X  (0.336)2 


=  0.00937  in  foot  and  pound  units. 

When  suspended  from  the  wristpin  end,  it  is  evident  that  h  = 
9.315  in.  or  0.776  ft.,  while  t  was  found  to  be  0.539  sec.  giving  the 
value 


I  =  ftii)t  X  2.281  X  0.776  -  0.0709  X  (0.776)  2 
=  0.00940. 

The  average  of  these  is  0.0094  which  may  be  taken  as  the 
moment  of  inertia  about  the  center  of  gravity.  The  square  of 
the  radius  of  gyration  about  the  same  point  is 

/x         0.0094X32.16 


W 

or  k  =  0.36  ft. 


INDEX 


Absolute  motion,  26 
Acceleration    in    machinery,    Chap. 
XV,  277 

angular,  282 

bending  moment  due  to,  287 

connecting  rod,  292,  297,  301 

effect  m  crank  effort,  295-297 
on  i  arts,  283 

force  lequired  to  produce,  284 

forces  at  bearings  due  to,  300 
due  to,  283 

general  effects  of,  277 

graphical  construction,  280 

in  bodies,  278 

in  engine,  290,  291,  300 

in  rock  crusher,  287 

links,  282 

normal,  278 

piston,  291-297,  300 

piston-formula,  329 

points,  282 

stresses  in  parts  from,  286 

tangential,  278 

total,  280 

vibrations  due  to,  277 
Addendum  line,  76,  83 

circle,  76,  83 
Adjustment   of  governors,    rapidity 

of,  236 
Angle  of  approach,  77 

friction,  183 

obliquity,  79 

recess,  77 
Angular  acceleration,  282 

space  variation,  258 

velocity,  35,  38,  39,  57 
Annular  gears,  80 
Approach,  arc  of,  77 

angle  of,  77 


Arc  of  approach,  77 

contact,  77 

recess,  77 
Automobile  differential  gear,  133 

gear  box,  115 
Available  energy  variations,  164 

B 

Balancing,  Chap.  XVI,  307 

connecting  rod,  320 

crank  angles  for,  322 

disturbing  forces,  315 

duplication  of  parts  for,  320 

four-crank  engine,  322 

general  discussion,  307 

locomotives,  323 

masses,  general,  318 

motor  cycle,  325 

multi-cylinder  engines,  320,  etc. 

primary,  316 

reciprocating  masses,  314,  320 

rotating  masses,  308 

short  connecting  rod,  316 

secondary,  318 

swinging  masses,  313 

weights,  316 
Base  circle,  77 
Beam  engine,  154 
Bearing,  15 

Belliss  and  Morcom  governor,  223- 
227 

curves  of,  227 

Bending   moment   due   to   accelera- 
tion, 287 
Bevel  gears,  Chap.  VI,  90 

cone  distance  for,  93 

teeth  of,  92 
Bicycle,  6 

Brown  and  Sharpe  gear  system,  84 
Buckeye  governor,  233 

curves  for,  236 


333 


334 


INDEX 


Cams,  Chap.  VIII,  136 

gas  engine,  143 

general  solution  of  problem,  144 

kinds  of,  137 

purpose  of,  136 

shear,  141 

stamp  mill,   137 

uniform  velocity,  140 
Center,  fixed,  30 

instantaneous,  28 

permanent,  30 

pressure,  190 

theorem  of  three,  30 

virtual,  28 
Chain  closure,  11 

compound,  17 

double  slider  crank,  22 

inversion  of,  17 

kinematic,  16 

simple,  17 

slider  crank,  18 
Characteristic    curve   for   governor, 

212 

Chuck,  elliptic,  22 
Circle,  base,  77 

describing,  72 

friction,  191 

pitch,  70 
Circular  or  circumferential  pitch,  77, 

84 

Clearance  of  gear  teeth,  83 
Cleveland  drill,  129 
Clock  train  of  gears,  114 
Closure,  chain,  11 

force,  11 
Coefficient  of  friction,  180 

speed  fluctuation,  265 
Collars,  10 
Compound  chain,  16 

engine,  170 

gear  train,  110 
Cone,  back,  distance,  93 

pitch,  92 
Connecting  rod,  4 

acceleration  of,  292 

balancing  of,  320 

friction  of,  193 


Connecting  rod,  velocity  of,  61 
Constrained  motion,  6,  10 
Contact,  arc  of,  in  gears,  77 

line  of,  skew  bevel  gears,  90 

path  of,  in  gears,  77 
Continued  fractions,  application,  122 
Cotter  design,  184 
Coupling,  Oldham's,  22 
Crank,  4 

Crank  angles  for  balancing,  322 
Crank  effort,  152,  165 

diagrams,  Chap.  X,  166 

effect  of  acceleration  on,   295, 

299 

of  connecting  rod,  297 
of  piston,  295 
Crankpin,  4 

Crossed  arm  governor,  206 
Crosshead,  friction  in,  183 
Crusher,  rock,  158 
Cut  teeth  in  gears,  83 
Cycloidal  curves,  73,  80 

teeth,  72 

how  drawn,  74 
Cylinders,  pitch,  70 


Dedendum  line,  83 
Describing  circle,  72,  75 
Diagrams  of  crank  effort,  166 

E-J  (energy-inertia),  262 

indicator,  167,  etc. 

motion,  Chap.  IV,  49 

polar,  45 

straight  base,  45 

torque,  166,  169 

vector,  phorograph,  58 

velocity,  uses  of,  49 

Chap.  Ill,  35 
Diametral  pitch,  84 
Differential  gear,  automobile,  133 
Direction  of  motion,  29,  30,  32 
Discharge,  pump,  46 
Divisions  of  machine  study,  8 
Drill  with  planetary  gear,  129 
Drives,  forms  of,  68 


INDEX 


335 


E 


Efficiency,  engine,  196 

governor,  194 

maqhine,  Chap.  XI,  177 

mechanical,  177 

shaper,  186 
Effort,  crank,  152,  165 
E-J  (energy-inertia)  diagram,  262 

for  steam  engine,  270 
Elements,  11 
Elliptic  chuck,  22 
Energy  available,  164 

kinetic  of  bodies,  243 
of  engine,  246 
of  machine,  244 

producing  speed  variations,  251 
Engine,  acceleration  in,  290,  291,  300 

beam,  154 

compound,  170 
'  crank  effort  in  steam,  166,  etc. 

diagram  of  speed  variation,  250, 
252,  255 

efficiency,  196 

energy,  kinetic,  246 

internal  combustion,  171 

multi-cylinder,  321,  etc. 

oscillating,  25 

proper  flywheel  for  steam,  270, 

274 

for  gas,  274 
Epicyclic  gear  train,  110,  124,  etc. 

ratio,  110,  125 
Epicycloidal  curve,  73 
Equilibrium  of  machines,  150 

static,  8 
External  forces,  149 


Face  of  gear,  width  of,  84 

tooth,  84 

Factor,  friction,  181 
Feather,  11 
Fixed  center,  30 
Fluctuations  of  speed  in  machinery, 

Chap.  XIII,  240 
approximate  determination  of, 
249 


Fluctuations  of  speed,  cause  of,  242 

conditions  affecting,  247,  260 

diagram  of,  255 

energy,  effect  on,  240,  247,  251 

in  any  machine,  248 

in  engine,  250,  252 

nature  of,  240 

Flywheels,   weight  of,   Chap.   XIV, 
261 

best  speed,  267 

effect  of  power  on,  261 
of  load  on,  261 

gas  engine,  274 

general  discussion,  262 

given  engine,  270 

minimum  mean  speed,  269 

purpose,  261 

speed,  effect  on,  264,  267 
Follower  for  cam,  138 
Forces  in  machines,  Chap.  IX,  149 

accelerating,  283,  300 
effect  on  bearings,  300 
general  effects,  283 

causing  vibrations,  314 

closure,  17 

external,  149 

in  machine,  151 

in  shear,  152 
Ford  transmission,  131 
Forms  of  drives,  68 
Frame,  3,  6 
Friction,  178 

angle  of,  183 

circle,  191 

coefficient  of,  180 

crosshead,  183 

factor,  181 

in  connecting  rod,  193 

in  cotter,  184 

in  governor,  194,  216 

laws  of,  180 

sliding  pairs,  181 

turning  pairs,  189,  192 


Gas  engine  cam,  143 
crank  effort,  172 
flywheel,  274 


336 


INDEX 


Gears  and  gearing,  Chap.  V  and  VI, 

68,  90 
Gears,  annular,  80 

bevel,  Chap.  VI,  90,  94 

Brown  and  Sharpe  system,  84 

conditions  to  be  fulfilled  in,  71 
when  used,  68 

diameter  of,  69 

examples,  85 

face,  84 

hunting  tooth,  123 

hyperboloidal,  90,  94 

interference  of  teeth,  81 

internal,  80 

methods  of  making,  83 

path  of  contact  of  teeth,  73 

proportions  of  teeth,  84 

sets  of,  79 

sizes  of,  69 

spiral,  Chap.  VI,  90 

spur,  68 

systems,  discussion  on,  87 

toothed,  Chap.  V,  68 

types  of,  90 

Gears  for  nonparallel  shafts,  91 
Gears,  mitre,  91 

screw,  90,  102 

skew  bevel,  90,  93 

spiral,  90,  102 
Gears,  worm,  90,  etc. 

construction,  104 

ratio  of,  103 

screw,  106,  107 
Gearing,  trains  of,  Chap.  VII,  110 

compound,  110 

definition  of,  110 

epicyclic,  110,  124,  etc. 

kinds  of,  110 

planetary,  see  Epicyclic. 

reverted,  110 

Gleason  spiral  bevel  gears,  93 
Gnome  motor,  20 

acceleration  in,  303 
Governors,  Chap.  XII,  201 

Belliss  and  Morcom,  223 
curves  for,  227 

Buckeye,  233 

characteristic  curves,  212 

crossed  arm,  206 


Governors,  definition,  201 

design,  219 

efficiency  of,  194 

friction  in,  194,  216 

Hartnell,  221 

design  of  spring  for,  222 

height,  205 

horizontal  spindle,  223 

inertia,  228,  229 
properties  of,  229 

isochronism,  206,  213,  230 

McEwen,  233 

pendulum,  theory  of,  203,  204, 
etc. 

Porter,  207 

powerfulness,  211,  212,  215 

Proell,  159,  220 

Rites,  238 

Robb,  230 

sensitiveness,  210,  214 

spring,  221 
design  of,  222 

stability,  207,  213 

types  of,  201,  202 

weighted,  see  Porter  governor. 
Governing,  methods  of,  201 
Graphical  representation,  see  Matter 
desired. 


II 


Hartnell  governor,  221 
Height  of  governor,  205 
Helical  motion,  9 

teeth,  87 

uses  of,  88 

Hendey-Norton  lathe,  119 
Higher  pair,  14 
Hunting  in  governors,  207 

tooth  gears,  123 
Hyperboloidal  gears,  91,  94 

pitch  surfaces  of,  95,  101,  108 

teeth  of,  102 
Hypocycloidal  curve,  73 


1 


Idler,  113 
Image,  53,  55 


INDEX 


337 


Image,  angular  velocity  from,  57 

copy  of  link,  59 

how  found,  55 

of  point,  53 

Inertia-energy  (E-J)  diagram,  262 
Inertia  governor,  203,  228,  etc. 

analysis  of,  233 

distribution  of  weight  in,  237 

isochronism  in,  234 

moment  curves  for,  234 

properties  of,  229 

stability,  234,  235 

work  done  by,  236 
Inertia  of  body,  331 

parts,  Chap.  XVI,  307 

reduced,  for  machine,  244 
Input  work,  176,  251 
Instantaneous  center,  28,  32,  35 
Interference  of  gear  teeth,  81 
Internal  gears,  80 
Inversion  of  chain,  18 
Involute  curves,  method  of  drawing, 
78,  79 

teeth,  78 

Isochronism  in  governors,  206,  213, 
230 


Jack,  lifting,  185 
Joy  valve  gear,  41 

velocity  of  valve,  41,  66 


K 


Kinematic  chain,  16 
Kinematics  of  machinery,  8 
Kinetic  energy  of  bodies,  243 

of  engine,  246 

of  machine,  244 


Lathe,  4;  116,  etc. 

Hendey-Norton,  119 
screw  cutting,  116 

Line  of  contact  in  gears,  90 

Linear  velocities,  29,  35 

Link,  16 


Link,  acceleration  of,  282 
angular  velocity  of,  68 
kinetic  energy  of,  243 
motion,  Stephenson,  63 
primary,  66 
reference,  49 
velocity  of,  36,  37,  40 
general  propositions,  38 

Load,  effect  on  flywheel  weight,  261, 
273 

Locomotive  balancing,  323 

Lower  pair,  14 


M 


Machine,  definition,  7 

design,  8 

efficiency  of,  Chap.  XI,  176 

equilibrium  of,  150 

forces  in,  Chap.  IX,  150 

general  discussion,  3 

imperfect,  8 

kinetic  energy  of,  244 

nature  of,  3 

parts  of,  5 

purpose  of,  7 

reduced  inertia  of,  244 

simple,  17 

Machinery,  fluctuations  of  speed  in 
Chap.  XIII,  240 

cause  of  speed  variations,  240 

effect  of  load,  etc.,  240,  247 

kinematics  of,  8 
McEwen  governor,  233 
Mechanical  efficiency,  177 
Mechanism,  17,  55 
Mitre  gears,  91 
Module,  85 
Motion,  absolute,  26 

constrained,  6 

diagram,  Chap.  IV,  49 

direction  of,  29,  30 

helical,  9 

in  machines,  Chap.  II,  24 

plane,  4,  9,  24 

quick-return,  20,  62 

relative,  5,  25,  26 

propositions  on,  26,  27 

screw,  9 


338 


INDEX 


Motion,  sliding,  11 

spheric,  9 

standard  of  comparison,  25 

translation,  3 

turning,  10 

Motor  cycle  balancing,  325 
Multi-cylinder  engines,  320,  322,  325, 
etc. 


N 


Normal  acceleration,  278 
Numerical    examples;     see    Special 
subject. 


Obliquity,  angle  of,  79 
Oldham's  coupling,  22 
Oscillating  engine,  19 
Output  work,  176,  251 


Pair,  3 

friction  in,  167,  170 

higher,  14 

lower,  14 

sliding,  12,  33 
friction  in,  181 

turning,  11 

friction  in,  189 
Permanent  center,  30 
Phorograph,  50,  52,  58,  Chap.  IV 

for  mechanism,  55 

forces  in  machines  by,  155 

principles  of,  50-53 

property  of,  66 

vector  velocity  diagram,  58 
Pinion,  69 
Piston,  3 

acceleration  of,  291-296,  329 

velocity,  44,  61 

Pitch,  circular  or  circumferential  77, 
84 

circle,  70 

cones,  92 

cylinder,  70 

diametral,  84 


Pitch,  normal,  101 

point,  84 

surfaces,  95,  100 
Plane  motion,  4,  9,  24 
Planetary  gear  train,  see  Epicyclic 
train. 

examples,  126 

purpose,  124 

ratio,  125 
Point,  acceleration  of,  282 

image  of,  53 

of  gear  tooth,  84 
Polar  diagram,  45 

Porter  governor,  see  also   Weighted 
governor. 

advantages,  209 

description,  207 

design  of,  219 

height,  209 

lift,  210 

sensitiveness,  210 

Power,  effect  on  flywheel  weight,  273 
Powerfulness  in  governors,  211,  215 
Pressure,  center  of,  190 
Primary  balancing,  316 
Primary  link,  56 
Proell  governor,  159,  220 
Pump  discharge,  46 


Quick-return  motion,  20,  62,  186 
Whitworth,  20,  62 


II 


Racing  in  governors,  207 

Rack,  80 

Rapidity  of  adjustment  in  governors, 

236 
Recess,  angle  of,  77 

arc  of,  77 

Reciprocating  masses,  balancing  of, 
314,  320 

balancing  by  duplication,  320 
Reduced  inertia,  244 
Reeves  valve  gear,  64 
Reference  link,  49,  see  Primary  link. 
Relative  motion,  5,  25,  26 


INDEX 


339 


Relative  velocities,  38,  40 
Resistant  parts  of  machines,  5 
Reverted  gear  train,  110 
Rigid  parts  of  machines,  5 
Rites  governor,  238 
Riveters,  toggle  joint,  161 
Robb  governor,  230 
Rock  crusher,  158 

acceleration  in,  287 
Rocker  arm  valve  gear,  60 
Root  circle,  76,  83 

of  teeth,  84 
Rotating  masses,  balancing  of,  308 

pendulum  governor,  202,  etc. 
defects  of,  205 
theory  of,  204 
Rotation,  sense  of,  in  gears,  113 

sense  of,  for  links,  57 


Screw  gears,  90,  102 

motion,  9 

Secondary  balancing,  319 
Sensitiveness  in  governors,  210,  214 
Sets  of  gears,  79 

Shaft  governor,  203,   229;  see  also 
Inertia  governor. 

properties  of,  228 
Shaper,  efficiency  of,  187 
Shear,  cam,  141 

forces  in,  157 
Simple  chain,  17 
Skew  bevel  gears,  90,  94 

pitch  surfaces  of,  95 
Slider-crank  chain,  18 

double  chain,  22 
Sliding  motion,  11 

friction  in,  181 

pair,  12,  33 

Slipping  of  gear  teeth,  76 
Space  variation,  angular,  258 
Speed     fluctuations'    in    machines, 
Chap.  XIII,  240 

approximate  determination,  249 

cause  of,  240 

coefficient  of,  265 

conditions  affecting,  247,  260 

diagram  of,  255 


Speed  fluctuations,  effect  of  load  on, 
240,  247 

energy  causing,  251 

in  engine,  250,  252 

in  any  machine,  248 

minimum,  mean,  269 

nature  of,  240 

Speed  of  fly-wheel,  best,  267 
Spheric  motion,  9 
Spiral  bevel  teeth',  90,  93 

gears,  Chapter  VI,  90 
Spring  governor,  221 
Spur  gears,  68 

Stability  in  governors,  196,  207,  213 
Stamp-mill  cam,  137 
Static  equilibrium,  8,  150 
Stephenson  link  motion,  63 
Stresses  due  to  acceleration,  286 
Stroke,  4 
Stub  teeth,  79,  85 
Sun  and  planet  motion,  128 
Swinging  masses,  balancing  of,  313 


Tangential  acceleration,  278 
Teeth,  cut,  83 

cycloidal,  72,  74 

drawing  of,  74,  78 

face,  84 

flank,  84 

helical,  87 

hyperboloidal,  117 

interference,  81 

involute,  77,  79 

of  gear  wheels,  76 

parts  of,  84 

path  of  contact,  73,  78 

point,  84 

profiles  of,  73 

root,  83 

slipping,  amount  of,  76 

spiral  bevel,  93 

stub,  79,  85 

worm  and  worm-wheel,  103,  104 
Theorem  of  three  centers,  30 
Threads,  cutting  in  lathe,  122 
Three-throw  pump,  47 
Toggle-joint  riveters,  161 


340 


INDEX 


Toothed  gearing,  Chap.  V,  68 
Torque  on  crankshaft,  165,  etc. 

effect  of  acceleration  on,  299,  301 
Total  acceleration  of  point,  280 
Trains  of  gearing,  Chap.  VI,  90 

automobile  gear  box,  116,  133 

change  gears,  117 

clock,  114 

definition  of,  110 

epicyclic,  124 

examples,  114 

formula  for  ratio,  112 

lathe,  116 

planetary,  124 
ratio  of,  112 

rotation,  sense  of,  113 
Translation,  motion  of,  3 
Transmission,  Ford  automobile,  131 
Triplex  block,  Weston,  128 
Turning  motion,  10 

moment,  Chap.  X,  164 

pair,  11 

friction  in,  189 

U 

Uniform  velocity  cam,  140 
Unstable  governor,  207 


Valve  gears,  Joy,  41,  66 
Reeves,  64 
rocker  arm,  60 


Velocity,  diagram,  Chap.  Ill,  35 

graphical  representation,  43 

linear,  29,  35 
of  points,  35 
relative,  38,  40 

piston,  44 
Velocities,  angular,  35,  37,  39 

how  expressed,  39 
Vibration  due  to  acceleration,  277 
Virtual  center,  28,  32,  35 


W 


Watt  sun  and  planet  motion.  128 
Weight  of  fly-wheels,   Chap.   XIV, 
261 

effect  of  speed,  etc.,  on,  267 
Weighted  governor,  207 

advantages  of,  209 

effect  of  weights,  216 

height,  209 

lift,  210 

Weights  for  balancing,  316 
Weston  triplex  block,  128 
Wheel  teeth,  76 
Whitworth  quick-return  motion,  20, 

62 
Work,  7 

in  governors,  235 

input  and  output,  176 
Worm,  102 

gear,  102 

teeth  of,  105 
Wrist  pin,  4 


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